{"paper_meta":{"paper_id":"arxiv:0708.3568","title":"0708.3568","authors":[],"year":null,"published":null,"updated":null,"venue_or_source":"arxiv-papers-shard","primary_category":"cs","secondary_categories":[],"doi":null,"license":"unknown","source_type":"pdf+shard","source_url":null,"pdf_url":null},"abstract":{"raw":"","cleaned":"","offsets":[]},"full_paper_text":{"raw_ordered_text":"Vadim Tarin\n A Polynomial-time Algorithm for \nComputing the Permanent in GF( q\n3 ) \nAbstract: \nA polynomial-time algorithm for computing the permanent in any field of characteristic 3 is \npresented in this article. The principal objects utilized for that purpose are the Cauchy and \nVandermonde matrices, the discriminant function and their generalizations of various types. \nClassic theorems on the permanent such as the Binet-Minc identity and Borchadt's formula \nare widely applied, while a special new technique involving the notion of limit re-defined for \nfields of finite characteristics and corresponding computational methods was developed in \norder to deal with a number of polynomial-time reductions. All the constructions preserve a \nstrictly algebraic nature ignoring the structure of the basic field, while applying its infinite \nextensions for calculating limits.\nA natural corollary of the polynomial-time computability of the permanent in a field of a \ncharacteristic different from 2 is the non-uniform equality between the complexity classes \nP and NP what is equivalent to RP=NP( Ref. [1]).\n Unless specified otherwise, \n all the results are for fields of Characteristic 3.\nDefinitions and denotations.\nFor \nm\nn \n-matrices \nm\nn\nh\nij\ndef\nh\nm\nn\nij\nij\ndef\na\nA\nb\na\nB\nA\nB\nA\n \n \n \n \n \n \n \n}\n{\n,\n}\n{\n,\n,\n \n ,\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nq\nq\nh\nh\ndef\nh\nh\nA\nA\nA\n...\n1\n1\n)\n,...,\n(\n (for real numbers \nqh\nh\nh\n,...,\n,\n1\n)\n(the Hadamard product, degree and vector-degree),\nfor \n)\n,\n(\n}\n,...,\n1\n{\n},\n,...,\n1\n{\nJ\nI\nA\nm\nJ\nn\nI\n \n \n is the sub-matrix of A whose \nset of rows is I and of columns J ,\n\nfor matrices \n]\n[\n]\n1[ ,...,\ns\nA\nA\n with the same number of rows \ns\nr\nr\nA\n1\n]\n[\n \n denotes \nthe matrix \n \n]\n[\n]\n1\n[\n|\n...\n|\ns\nA\nA\n,\n \n \n \n \n \nn\ni\nij\nm\nj\na\nA\nscal\n1\n1\n)\n(\n ,\n \n;\ns)\nfor vector\n(as\n)\n('\n),\n(\nsimply \n write\n will\nwe\n1\n)\ndim(\nfor\n,)\n(\n...\n)\n('\n,)\nmatrix\ne\nVandermond\nthe\n(\n)\n(\n)\n(\n,\n)\n(\n]\n[\n]\n[\n]\n[\n)\ndim(\n1\n]\n[\n)]\n[dim(\n)\n1\n,...,\n0\n(\n]\n[\nt\nvan\nt\nvan\nt\nt\nVan\nt\nd\nd\nt\nd\nd\nt\nVan\nt\nVan\nt\nVan\nt\nt\nVan\nh\nh\nh\nt\ndef\nh\nt\ndef\nh\nT\ndef\nh\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n2\n(mod\n1\n)\ndim(\n,\n)\n(\n)\n(\n),\n2\n(mod\n0\n)\ndim(\n,\n)\n(\n)\n(\n)\n(\n]\n2\n/)\n1\n)\n[(dim(\n]\n2\n/)1\n)\n[(dim(\n]\n2\n/)\n[dim(\n]\n2\n/)\n[dim(\nt\nt\nVan\nt\nVan\nt\nt\nVan\nt\nVan\nt\nW\nt\nt\nt\nt\ndef\n,\n)\n(\n)\n,\n(\n)\ndim(\n1\n)\ndim(\n1\n \n \n \n \n \n \nu\ni\nv\nj\nj\ni\ndef\nv\nu\nv\nu\npol\nfor a scalar \n)\n,\n(\n)\n,\n(\"\n,)\n,\n(\n)\n,\n('\n,\n2\n2\nv\npol\nd\nd\nv\npol\nv\npol\nd\nd\nv\npol\ndef\ndef\n \n \n \n \n \n \n \n \n \n \n \n ;\nfor an appropriate set I , \nIu\nis the sub-vector of u consisting of the coordinates \nwith indexes from I (including the case when I is a multi-set, with correspondingly \nduplicated coordinates),\nI\nu\\\nis its compliment in u ;\nfor a scalar , \nm\nn \n \n is the \nm\nn \n-matrix all whose entries equal ,\nfor \n1\n \nm\nwe will write simply \nn\n ;`\nfor a set H , \n)\n(H\nPart\n is the set of its partitions on nonempty sets;\nfor a finite sequence d, |d| denotes the number of its members;\nfor a real number \n \n \n \n \n \n0\n,1\n0\n,0\n)\n(\n,\nu\nu\nu\nu \n ;\nwhere \n)\ndim(\n|\n|,\n1,\n|\n|,\n1,\n)\ndim(\n)\ndim(\n)}\n,...,\n,\n,...,\n{(\n)}\n,\n{(\n}\n{\na\ni\ni\ni\ni\na\ni\ni\na\ni\ni\ni\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n is a \nmatrix)\nextension \nCauchy \nthe\n(\n}\n}\n1\n)!\n)(\n!\n(\n1\n)){\n(\n1\n{(\n)\n,\n(\n);\nmatrix\nCauchy \nthe\n(\n}\n1\n{\n)\n,\n(\n)\ndim(\n)\ndim(\n|\n||\n|\n,\n,\n)\ndim(\n)\ndim(\n,\n,\n,\n,\na\na\nj\ni\nj\ni\nl\nj\nk\ni\ndef\ny\nx\nj\ni\ndef\nj\ni\nl\nj\nl\nj\nk\ni\nk\ni\na\na\na\nd\nd\na\nd\nd\nj\ni\na\nE\ny\nx\ny\nx\nC\n\n)\ndim(a -vector whose coordinates are pairs of non-decreasing nonnegative integer \nfinite sequences, including the empty one. The rows and columns containing the\nterm \nia will be called the \nia -extension (and \nia itself the extension’s value );\n)\n,\n(\ni\ni\ni\n \n \n \n we’ll call the extension-degree of \nia , where \ni is the left \nextension-degree, \ni\n is the right one, \n|\n|\ni \n is the extension’s height , \n|\n|\ni\n \nis its width , and \n|\n|\n|\n|\n)\n(\ni\ni\ndef\ni\nbal\n \n \n \n \n \n \n is its balance . If \n0\n|\n|\n|\n|\n \n \ni\ni\n \n \nthen we’ll say that the matrix has an \nia -singularity. Singularities with \n(0,0)\n \ni \nwill be called simple. The sum \n \n \n \n)\ndim(\n1\n)\n(\n)\n(\na\ni\ni\nbal\nbal\n \n \nwill be called the total balance\nof the extension-degrees’ vector .\n \nfield.\nbasic\n \nover the\nfunction \n-\nweight\nits\nis\n )\n(\ndegrees,\n-\nextension\nof\nset \na\nis\n \n),\ndim(\n1,...,\nfor \nwhere,\n))\n,\n(\n(\n))\n(\n(\n))\n(\n(\n...\n \nexpression\n the\nunderstand\nll\n we'\nsum\n-\nE\nan \nBy \n1\n1\n)\ndim(\n)\ndim(\n)\ndim(\n1\ni\ni\ni\na\ni\ni\ni\nweight\na\ni\na\nE\nper\nbal\nweight\na\na\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThe formal sum \n \n \n \n \ni\ni\ni\ni\ni\ni\nweight\n \n \n \n \n)\n(\n will be defined as the extension- plane\nof \nia , \n)\n(\ni\ni\nweight \n will also be called the \nweight\nplane -\n of \ni in \ni , the pair \n)\n,\n(\ni\nia \nwill be called an extension-variety and thus we can also talk about the E-sum of \na on the extension-planes \n),\ndim(\n1,...,\n \n,\na\ni\ni\n \n \n while the pair of vectors \n)\n,\n(\n \na\nwill be called a vector-variety. If \ni has the form \n)\n0,\n(\n)\n,\n(\n \n \n \n \ni \n then we’ll say \nthat the E-sum is right-prolonged on \nia , if \n)\n,0\n(\n)\n,\n(\n \n \n \n \ni \n then left-prolonged,\nand the corresponding patterns of \n)\n,\n(\n \na\n (i. e. sub-vectors of the vector-variety) will be \ncalled right and left prolongations (correspondingly).\nWe will denote the E-sum of a on by \n)\n,\n(\n \na\nesum\n (we’ll use the brackets\n \n only \nfor left and right extension-degrees whose cardinalities exceed 1);\n)\n)\n0,0\n(,\n(\n)\n(\n~\n),\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n)\n,\n,\n(\n~\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\nz\ndef\nz\ny\nx\ndef\nz\nE\nz\nC\nz\ny\nx\nE\nz\ny\nx\nC\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n ;\nfor an \nr\nn -matrix A and an\nr\nm -matrix B ,\nm\nn\nm\n,\n2\n \n is even :\n(the copermanent of A with B ) ,\n)\n(\n)\n(\n)\n,\n(\n})\n,\n{\n},\n,...,\n1\n({\n)\n},\n,...,\n1\n({\n2\n/\n|\n|,\n|\n|,\n:\n,\nJ\nJ\nm\nI\nn\nm\nJ\nn\nI\nJ\nI\nJ\nI\ndef\nB\nper\nA\nper\nB\nA\ncoper\n\n,\n)\n(\n)\n(\n,\n)\ndet(\n)\ndet(\n|\n|,\n)\n},\n,...,\n1\n({\n|\n|,\n)\n},\n,...,\n1\n({\n \n \n \n \n \n \nn\nJ\nJ\nJ\nn\nn\nJ\nJ\nJ\nn\nA\nper\nA\nper\nA\nA\n \n \n \n \n \n \n \nm\nk\nJ\nj\nkj\nJ\nn\nn\nJ\nJ\ndef\nB\nb\nA\nper\nA\nr\ne\np\n1\n)\n},\n,...,\n1\n({\n|\n|,\n)\n(\n)\n(\nˆ\n(the permanent of A on the base B , or the base-permanent of A on B );\nfor r-vectors \n \n ,\n,u\n, nonnegative integers \nq\np \n:\n \n \n \n \n \n \n \n \n \n \nI\ni\nJ\nj\nj\ni\nJ\nI\nq\nJ\np\nI\nJ\nI\nJ\nI\ndef\nq\np\nu\nVan\nu\nVan\nu\ndis\n)\n)(\n))(\n(\n(\ndet\n))\n(\n(\ndet\n)\n,\n,\n(\n2\n2\n|\n|,\n|\n|,\n:\n,\n,\n \n \n \n \n(the \nq\np, -discriminantal of u on \n \n ,\n) ;\nthe E-generated functions (for vectors with appropriate dimensions):\nthe star-function:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n2\n/)\ndim(\n|\n|,\n,\n,\n}\n,\n,\n{\n\\\n]\n2\n[\n]\n1\n[\n]\n2\n[\n]\n1\n[\n))\n(\n)\n,\n(\n(\n))\n,\n(\n(\n)\n)(\n(\n)\n,\n,\n,\n,\n(\nz\nJ\nI\nJ\nI\nJ\nI\nI\ni\nJ\nJ\nI\nJ\nI\nJ\nI\nJ\nj\nj\ni\ndef\ndef\nDiag\nu\nz\nC\nper\nz\nz\nC\nper\nu\nz\ngen\n \n \n \n \n \n \nthe 2-waves-function:\n)\n)\n0,0\n(\n,\n(\n)\n)\n0,0\n(\n)\n0,\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n)\n,\n,\n,\n,\n,\n(\n|\n|\n)\ndim(\n\\\n\\\n\\\n|\n|\n|\n|\n)\ndim(\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nw\nI\nJ\nI\nI\nJ\nJ\nI\ny\nx\nJ\nI\ndef\ndef\nw\nz\nesum\nw\nz\nt\nh\nesum\nt\nh\nw\nz\ngen\n \n \n \n \n)\n0,1(\n)1,0\n(\n)\n0,0\n,\n0,0\n(\n \n where\n \n \n \n \n \n \n \n \n \n \n \n \n \n \ndef\n.\nWe’ll call the extension-plane (0,0) the wave and the biwave; together with \ntheir values (say \niz and \nj\nw ) such extension-varieties will be called a \niz -wave \nand a \nj\nw -biwave (correspondingly) ; \nthe wave-function (a partial case of the 2-waves-function):\n;\n))\n(\n~\n(\n)\n))(\n,\n,\n(\n~\n(\n)\n,\n,\n,\n,\n,\n(\n)\n,\n,\n,\n,\n(\n)\ndim(\n|\n|,\n\\\n~\n \n \n \n \n \n \n \n \n \n \n \nI\nh\nJ\nJ\nJ\nj\nj\nI\nI\ni\ni\nI\nJ\ndef\nz\nC\nper\nz\nt\nh\nC\nper\nt\nh\nz\ngen\nt\nh\nz\ngen\n \n \n \n \n \n \nFor both the wave-function and the 2-waves-function \ni\n will be called the i -th ebb, \nand the ebb vector.\nthe base-function:\n\nI\nt\nu\nC\nI\ni\nh\ni\nh\nh\nt\nh\nh\nt\nu\nC\nh\nu\nC\ndef\nDiag\nt\nh\nC\nr\ne\np\nh\nu\nC\nDiag\nt\nh\nC\nI\nr\ne\np\nt\nh\nu\ngen\nI\n))\n(\n)\n,\n(\n(\nˆ\n))\n1\n)\n,\n(\n(\n(\n)\n)\n(\n)\n,\n(\n0\n0\n(\nˆ\n)\n,\n,\n,\n(\n)\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\n,\n(\n)\n,\n(\n^\n\\\n \n \n \n \n \n \nif F is a basic field and is a formal variable then for\n \n \n \n \n \n \n \n \n \n \n \n \n0\n)\n(\n,\nexist\nt \ndoesn'\n0\n)\n(\n,\n)\n(\nlim\n,\n)\n(\n:)\n(\n]\n0\n[\n0\n]\n[\ng\norder\ng\norder\ng\ng\nq\ng\norder\nF\ng\ng\nq\nk\nk\nk\n \n \n \n \n \n \n \nBy \n)\n),\n(\n(\nx\nx\nf\n \n we’ll denote the Jacobian matrix \n)\ndim(\n)\ndim(\n}\n)\n(\n{\nx\nf\nj\ni\nx\nx\nf\n \n \n \n .\nPreliminaries.\nThe Binet-Minc identity (for any characteristic, Ref. [3], [4]):\nLet A be an \n \n m\nn\nmatrix, then\n)\n(\n})\n,...,\n1\n({\n1\n)!\n1\n|\n(|\n)1\n(\n)\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \nn\nPart\nI\nm\nj\nI\ni\nij\nn\na\nI\nA\nper\n,\nThe Binet-Minc identity for Characteristic 3:\nLet A be an \n \n r\nn\nmatrix, then\n \n \n \n \n \n \n \n \n \n \n})\n,...,\n1\n({\n)\n,\n(\n3\n,\n2\n,1\n)\n(\n)\n(\n)!\n1\n|\n(|\n)1\n(\n)\n(\nn\nPart\nI\nR\nI\nn\nA\nscal\nI\nA\nper\n ,\nwhere:\n)\n(\n3,2\n,1\nH\nPart\n is the set of partitions of the set H on nonempty sets of cardinalities \nnot bigger than 3, \n}\n,...,\n1\n{\nr\nR \n.\nFor our purposes we’ll also use this identity in the form \n \n \n \n \n \n \n \n \n \n \n})\n,...,\n1\n({\n)\n,\n(\n|\n|\n1\n3\n,\n2\n,1\n)\n(\n)\n(\n)!\n1\n|\n(|\n)1\n(\n)\n(\nn\nPart\nI\nR\nI\nI\nA\nscal\nI\nA\nper\nThe generalized Binet-Minc identity for \n)\n,\n(\nB\nA\ncoper\n :\nLet A , B be an \nr\nn - and \nr\nm -matrix, \n)\n2\n(mod\n0\n \nm\n, then\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nK\nn\nN\nK\nm\nA\nscal\nI\nB\nA\nscal\nB\nA\ncoper\n)\n(\n)\n(\n)\n(\n)\n)!\n1\n|\n(|\n(\n)\n(\n)1\n(\n)1\n(\n)\n,\n(\n)\n,\n(\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|\n|\n2\n/\n3\n,\n2\n,1\n2\n;\n2\n,1\n\nwhere:\n},\n,...,\n1\n{\n},\n,...,\n1\n{\n},\n,...,\n1\n{\nr\nR\nm\nM\nn\nN\n \n \n \nfor two disjoint sets \n'\nH and \n\"\nH ,\n)\"\n,'\n(\n2\n;\n2\n,1\nH\nH\nPart\n is the set of partitions of the set \n\"\n' H\nH \non sets having 1 or 2 elements in \n'\nH and 2 elements in \n\"\nH (those sets are denoted here as \npairs \n\"\n\"\n,'\n'\n),\n\"\n,'\n(\nH\nI\nH\nI\nI\nI\n \n \n).\nProof:\nthe right part can be re-written as \n)\n(\n)\n(\n)\n(\n)1\n(\n)!\n1\n|\n(|\n)\n(\n)\n,\n(\n1\n|\n|\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n3\n,\n2\n,1\n2\n;\n2\n,1\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nN\nK\nA\nscal\nI\nB\nA\nscal\n. \nLet’s denote it by \n)\n,\n(\nB\nA\n \n.\nThen, by the construction,\n)\n(\n}\n,...,\n{\n)\n(\n}\n,...,\n{\n}\n,...,\n1\n{\n,\n,\n'\n\"\n\"\n'\n|\n|\n|\n|\n\\\n\"\n|\n|\n\"\n1\n'\n|\n|\n'\n1\n\"\n1\n'\n1\n\"\n|\n|\n'\n|\n|\n'\n\"\n\"\n'\n)\n1,\n1\n(\n...\n)\n,\n(\nM\nPart\nI\nI\nN\nPart\nI\nI\nr\nJ\nI\nI\nI\nI\nJ\nj\nI\ni\nI\ni\nj\ni\nj\ni\nI\nI\nJ\nJ\nJ\nJ\nj\nj\nj\nj\nb\na\nB\nA\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere \n \n\\}\n{\n means the set of sets with removed -s.\nThen let's prove that \n))\n2\n(\n)\n(\n)(\n1\n(\n)\n1,\n1\n(\n \n \n \n \ns\ns\nq\ns\nq\n \n \n \n \n \n \n .\nThe generalized Binet-Minc identity for \n)\n(\nˆ\nA\nB\nr\ne\np\n:\nLet A , B be an \nr\nn - and \nr\nm -matrix, then\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nI\nI\nK\nn\nN\nK\nB\nA\nscal\nI\nB\nA\nscal\nA\nr\ne\np\n)\n(\n)\n(\n)\n(\n)\n)!\n1\n|\n(|\n(\n)\n(\n)1\n(\n)1\n(\n)\n(\nˆ\n)\n,\n(\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|)\n\"\n|\n1\n)(\n2\n|'\n(|\n|\n|\n3\n,\n2\n,1\n1\n;\n2\n,1\n)\n(\n \nwhere:\n},\n,...,\n1\n{\n},\n,...,\n1\n{\n},\n,...,\n1\n{\nr\nR\nm\nM\nn\nN\n \n \n \nfor two disjoint sets \n'\nH and \n\"\nH ,\n)\"\n,'\n(\n1\n;2\n,1\nH\nH\nPart\n \n is the set of partitions of the set \n\"\n' H\nH \non sets having 1 or 2 elements in \n'\nH and at least 1 element in \n\"\nH (those sets are denoted \nhere as pairs \n\"\n\"\n,'\n'\n),\n\"\n,'\n(\nH\nI\nH\nI\nI\nI\n \n \n).\nProof:\nby analogy with the previous proof, we re-write the right part as\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nI\nI\nN\nK\nA\nscal\nI\nB\nA\nscal\n)\n(\n)\n(\n)\n(\n)1\n(\n)!\n1\n|\n(|\n)\n(\n)1\n(\n)\n,\n(\n1\n|\n|\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|)\n\"\n|\n1\n)(\n2\n|'\n(|\n)\n(\n3\n,\n2\n,1\n1\n;\n2\n,1\n \nLet’s denote it by \n)\n,\n(ˆ\nB\nA\n \n.\n\nThen, by the construction,\n)\n(\n}\n,...,\n{\n)\n(\n}\n,...,\n{\n}\n,...,\n1\n{\n,\n,\n'\n\"\n\"\n'\n|\n|\n|\n|\n\\\n\"\n|\n|\n\"\n1\n'\n|\n|\n'\n1\n\"\n1\n'\n1\n\"\n|\n|\n'\n|\n|\n'\n\"\n\"\n'\n)\n1,\n1\n(ˆ\n...\n)\n,\n(ˆ\nM\nPart\nI\nI\nN\nPart\nI\nI\nr\nJ\nI\nI\nI\nI\nJ\nj\nI\ni\nI\ni\nj\ni\nj\ni\nI\nI\nJ\nJ\nJ\nJ\nj\nj\nj\nj\nb\na\nB\nA\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThen let's prove that \n)1\n(\n)\n1,\n1\n(ˆ\n \n \nq\ns\nq\n \n \n \n \nLemma 1 (known, for any characteristic, Ref. [2], [6]):\nLet \n).\ndim(\n)\ndim(\ny\nx \n Then\n)\n,\n(\n))\n(\ndet(\n))\n(\ndet(\n)1\n(\n))\n,\n(\ndet(\n2\n)\n1\n)\n)(dim(\ndim(\ny\nx\npol\ny\nVan\nx\nVan\ny\nx\nC\nx\nx\n \n \n \nCorollary 1.1 :\nLet \n).\ndim(\n)\ndim(\n2\ny\nx \n Then\n)\n,\n(\n))\n(\ndet(\n))\n(\n(\ndet\n)\n)\n,\n(\n)\n,\n(\ndet(\n2\n4\n2\ny\nx\npol\ny\nVan\nx\nVan\ny\nx\nC\ny\nx\nC\n \n \n \n \n \n \n \n \nProof:\n)\n(\n)\n(\n))\n,\n1\n(\ndet(\n)1\n(\nlim\n))\n)\n,\n1\n(\n)\n,\n(\ndet(\nlim\n)\n)\n,\n(\n)\n,\n(\ndet(\n)\ndim(\n)\ndim(\n)\ndim(\n0\n)\ndim(\n)\ndim(\n0\n2\ny\nx\nx\nC\ny\nx\nC\ny\nx\nC\ny\nx\nC\ny\nx\nC\nx\nx\nx\nx\nx\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nLemma 2 (for any characteristic, Ref. [5]):\nLet A , B be \nm\nn \n-matrices, \nm\nn \n. Then \n(*) \nFor \n)\n2\n(mod\n0\n \nn\n: \n)\n)}\n(\n{\ndet(\n)\n(\ndet2\nT\nm\nm\nA\nj\ni\nsign\nA\nA\n \n \n \n, \n)\n)}\n(\n({\n)\n)}\n(\n{\n(\n)\ndet(\nn\nn\nT\nm\nm\nj\ni\nsign\nPf\nA\nj\ni\nsign\nA\nPf\nA\n \n \n \n \n \nwhere for a real \n \n \n \n \n \n \n \n \n0\n,0\n0\n,1\n0\n,1\n)\n(\nu\nu\nu\nu\nsign\nu\n\nFor \n,\n,...,\n1\n,\nm\nq\nn\n \n \nscalar :\n)\ndet(\n)\n0\n0\n0\n2\n1\n1\n0\n0\n0\ndet(\n2\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n2\n2\n)\n2\n(\n2\nA\nI\nI\nA\nq\nm\nq\nm\nq\nq\nm\nq\nm\nq\nq\nm\nq\nq\nq\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n. \nIn the generic case this method allows to reduce A to a triangular matrix\nwhen computing \n)\ndet(A .\n(**) for \n)\n2\n(mod\n0\n \nn\n \n \n \n )\n|\n|\ndet(\n)\n|\ndet(\n1\n)\n,\n(\n)\n,\n(\n)\n,\n(\n)\n,\n(\n2\n/\n|\n|,\n)\n,\n(\n)\n,\n(\n \n \nm\nk\nk\nN\nk\nN\nk\nN\nk\nN\nn\nJ\nJ\nJ\nj\nj\nN\nj\nN\nB\nA\nB\nA\nB\nA\n \n \n \n \n \n \n \n ,\nwhere \n}\n,...,\n1\n{\nn\nN \n.\nLemma 3 (for any characteristic):\nLet \n)\n(x\nf\n be a polynomial in \n)\ndim(\n1,...,\nx\nx\nx\n of degree d . Then \n)))\n(\n(\n(\n)\n(\n0\n \n \n \n \n \n \n \n \n \nx\nx\nf\ncoef\nf\nd\ni\ni\nwhere \ni\ncoef is the coefficient at \ni (if the expression is considered as a polynomial in ).\nThe neighbouring computation principle (for any characteristic):\nLet \n)\n(u\nf\n be a polynomial in \n)\ndim(\n1,...,\nu\nu\nu\n of degree d over the field F , \n)\n,...,\n(\n)\ndim(\n1\nh\nh\nh\nu\nu \n , u exists at \n]\n0\n[h\n;\n)\n,...,\n(\n)\ndim(\n1\nh\nh\nh\nv\nv \n, \n)\ndim(\n]\n0\n[\n0\n)\n(\nv\nh\nv\n \n \n;\n)\n,\n(\n]\n0\n[h\nv\nu\n \n \n \n \n \n \n \n exists and is nonsingular. \nThen, given , over \n)\n( \nF\n: \n)))\n(\n(\n(\n)\n(\n0\n]\n[\n0\n \n \n \n \n \nd\nk\nk\nk\nd\ni\nh\nu\nf\ncoef\nf\ni\n \n \n \n.\nwhere:\nd\nk\nh\nu\ncoef\nh\nu\nk\nh\nh\nv\nu\nv\nk\ni\ni\ni\nk\nk\n,...,\n1\n,\n0\n))\n(\n(\n)\n)\n(\n)(\n1\n(\n)\n,\n(\n)\ndim(\n1\n0\n]\n[\n]\n0\n[\n]\n[\n]\n0\n[\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(And thus \n)\n(\n)\n)\n(\n(\n)\n(\n)\n(\n1\n]\n0\n[\n]\n0\n[\n0\n]\n[\n \n \n \n \n \n \n \n \nd\nd\nk\nk\nk\nO\nh\nu\nh\nu\nh\nu\n \n \n \n \n )\nThis method will be called the \nng\nneighbouri\n computation of the function \n)\n( \nf\n by the\nzation\nparameteri\n)\n(h\nu\n in the region\n)\ndim(\n0\n)\n(\nv\nh\nv\n \n \n from the \nbearing point \n]\n0\n[h\n.\n\nIf \n))\n(\n(\nh\nu\nf\n is computable in polynomial time for any h in the region\n)\ndim(\n0\n)\n(\nv\nh\nv\n \n \n and there exists a bearing point, then \n)\n( \nf\n is computable \nin polynomial time for any too. \nIn the further, to be clear and short, we’ll call a system of functions S \nalgebraically absolutely independent in a region R (given by a system \nof equations with a zero right part) iff the joint system of functions consisting \nof S and the left part of the system representing R is algebraically independent \nat some point of R. \nThe prolongation-derivative principle (for any characteristic):\na left (right) prolongation pattern of a vector-variety can be removed with an \ninduced coordinate-wise transformation of the remaining vector of extension-planes \nsuch that the E-sum preserves its value. Such a transformation will be called a left (right) \nprolongation-derivative, or the prolongation-derivative on the given prolongation. \nFormally it will be denoted (correspondingly for the left and right cases) by\n)\n)1,\n(\n)\n,\n(\n,\n(\n)\n)\n,\n,\n(\n,\n(\n)\n,\n(\n)\n,\n(\n)\n)\n,1(\n)\n,\n(\n,\n(\n)\n)\n,\n,\n(\n,\n(\n)\n,\n(\n)\n,\n(\n)\ndim(\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nesum\nd\nd\nesum\nesum\nd\nd\nesum\nd\nd\nesum\nesum\nd\nd\ndef\nR\nR\ndef\nR\nR\ndef\nL\nL\ndef\nL\nL\nThis principle is based on the Binet-Minc identity. \n \nLemma 4:\nLet A be an \nn\nn \n-matrix, B be a \nn\nn \n)\n2\n(\n-matrix. Then \n \n \n \n \n)\n(\ndet\n)\n|\n(\n)\n|\n(\n2 A\nB\nB\nper\nBA\nBA\nper\n \nThe Borchardt formula (for any characteristic, Ref. [2]) :\nLet \n)\ndim(\n)\ndim(\ny\nx \n. Then\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n))\n,\n(\n(\n2\ny\nx\nC\ny\nx\nC\ny\nx\nC\nper\n \n \nLemma 5:\n))\n(\ndet(\n)\ndet(\n))\n(\n(\n)\ndim(\nt\nVan\nt\nt\nW\nper\nt\n \n \n \nwhere \n)\ndim(t\n \nis the first \n)\ndim(t members of the sequence \ncomposed as the sequence of pairs \n \n \n \n,...,\n0\n),\n1\n3,\n3\n(\ns\ns\ns\n\nProof:\n \n \n \n \n \n \n)\ndim(\n1\n0\n))\n(\n(\n))\n,\n(\n(\nlim\n))\n(\n(\ny\nj\njt\nx\nW\nper\nt\nx\nC\nper\nt\nW\nper\n \n \n \nThe main part.\nTheorem I :\nLet \nn\nu\nt\n2\n)\ndim(\n2\n)\ndim(\n \n \n.\nThen\nProof:\n))\n(\n(\ndet\n)\n,\n(\n))\n(\n(\n)\n,\n(\n)\n)\n(\n0\n0\n)\n(\n)\n(\n(\n)\n,\n(\n))\n,\n(\n)\n)}\n,\n(\n({\n(\n))\n,\n(\n(\n2\n2\n1\n2\nu\nR\nu\nt\npol\nt\nW\nper\nu\nt\npol\nu\nR\nu\nR\nt\nW\nper\nu\nt\npol\nu\nt\nC\nu\nt\npol\nDiag\nper\nu\nt\nC\nper\nT\nT\nn\ni\ni\nn\ni\n \n \n \n \n \n \n \n \n \n \n \n \nwhere \n)\n(u\nR\n is the matrix of the coefficients of the polynomials\nin the formal scalar variable \n)\n,\n(\n),...,\n,\n(\n\\\n1\\\nn\nu\npol\nu\npol\n \n \n \n. The determinant \nof this matrix is a homogenous polynomial in \nn\nu\nu ,...,\n1\n of degree \n2\n)1\n(\n)1\n(\n...\n1\n0\n \n \n \n \n \n \nn\nn\nn\n, and in the meantime it is divided by \nj\ni\nu\nu \nfor any \nn\nj\ni\n \n \n \n1\n, or by \n))\n(\ndet(\nu\nVan\n; moreover, its coefficients \nare 1 and -1, therefore \n))\n(\n(\ndet\n))\n(\n(\ndet\n2\n2\nu\nVan\nu\nR\n \n, what completes \nthe proof.\nCorollary I.1 :\nLet \n)\ndim(\n)\ndim(\n2\n)\ndim(\nv\nu\nt\n \n \n.Then\n)\n,\n(\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n))\n,\n(\n(\n2\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nu\nt\npol\nv\nu\npol\nu\nVan\nv\nt\nW\nper\nv\nu\nu\nt\nC\nper\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n))\n,\n(\n(\n2\nu\nt\npol\nu\nVan\nt\nW\nper\nu\nu\nt\nC\nper\n\nProof:\n))\n,\n1\n(\n(\nlim\n))\n,\n(\n(\n)\ndim(\n)\ndim(\n0\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\nu\nu\nv\nt\nC\nper\nv\nu\nu\nt\nC\nper\nv\nv\n \n \n \n \nand, basing on Theorem I, we calculate the limit to get the lemma’s equality.\nCorollary I.2 :\nLet \n)\ndim(\n)\ndim(\n2\n)\ndim(\nv\nu\nt\n \n \n+1. Then\nProof:\n)\n/\n))\n/\n1\n,\n(\n(\n(\nlim\n)\n1\n)\n,\n(\n(\n0\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nu\nu\nt\nC\nper\nv\nu\nu\nt\nC\nper\nt\n \nand then, basing on Corollary I.1, we again just calculate the limit.\nTheorem II :\n)\n)\n(\n)\n(\n0\n)\n('\n)\n(\n)\n('\n)\n(\ndet(\n)\n,\n,\n(\n)\ndim(\n1\n]\n[\n]\n[\n]\n[\n]\n[\n]\n[\n]\n[\n,\n \n \nu\ni\ni\nq\np\ni\nq\np\nq\np\ni\nq\np\ni\ni\nq\np\ni\ni\nq\np\ni\ni\nq\np\ni\nq\np\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\ndis\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nProof:\nbasing on Lemma 2, we conclude that the right part of this equality is\n \n \n \n \n \n \n \n \n \n \n \n \n \n \np\nI\nI\nI\ni\ni\nq\np\nq\np\ni\nq\np\ni\ni\nq\np\ni\nu\nvan\nu\nvan\nu\nvan\n|\n|,\n]\n[\n]\n[\n]\n[\n)\n)\n(\n0\n)\n('\n)\n(\ndet( \n \n \n \nTheorem III :\nLet\n)\n,\n(\n)\n,\n('\n,0\n)\n,\n(\"\n,1\n3\n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n|\n|\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \ny\nx\npol\nt\npol\ny\nx\npol\nn\nt\nn\ny\nn\nx\nI\n \n \n \nThen\n)\n,\n(\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n)\n1\n)\n,\n(\n(\n2\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nu\nt\npol\nv\nu\npol\nu\nVan\nv\nt\nW\nper\nv\nu\nu\nt\nC\nper\nt\n\nI\ni\ni\nI\nI\ny\ny\npol\ny\nVan\nt\nx\npol\nx\nt\nW\nper\ny\nx\nC\nper\n)\n,\n('\n1\n))\n(\n(\ndet\n)\n,\n(\n))\n(\n(\n))\n,\n(\n(\n2\nProof:\nLet us apply the Binet-Minc identity to \n)\n)\n1\n|\n)\n,\n(\n((\n)\ndim(\nT\nt\nI\nI\ny\ny\nx\nt\nC\nper\n \n \n \n \n \n \n \n \n \n \n \nTheorem IV :\nLet\n)\n,\n(\n)\n,\n('\n,0\n)\n,\n(\"\n)\n2\n(mod\n0\n)\ndim(\n,1\n3\n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \ny\nx\npol\nt\npol\ny\nx\npol\nm\nz\nn\nt\nn\ny\nn\nx\n \n \n \nThen\n \n))\n(\n)\n,\n(\n),\n(\n)\n,\n(\n(\n \n \nDiag\ny\nz\nC\nDiag\ny\nx\nC\ncoper\n)\n}\n)\n,\n(\n{,\n}\n)\n,\n('\n{,\n(\n)\n,\n(\n))\n(\n(\n))\n(\n(\n2\n2\n2\n2\n/\n,\nn\ni\ni\nn\ni\ni\nm\nn\nz\ny\npol\ny\ny\npol\ny\ndis\nt\nx\npol\nz\nW\nper\nx\nt\nW\nper\n \n \n \n \n \n \n \n \n \nProof:\nIt follows from the definition of the copermanent and Theorems I, III. \n----------------------------------------------------------------------------------------\nLet’s note that for \n0\n)\n,\n(\"\n \n \n \n \n \n \n \ny\nx\npol \n to be true it’s sufficient that \n0\n)\n,\n(\"\n \n \n \n \n \n \n \ny\nx\npol \n for any n pair-wise distinct values of (due to the characteristic), \nfor instance for all the coordinates of x , i.e. it’s equivalent to \nn\nn\ni\ny\nx\nx\npol\n0\n)}\n,\n(\"\n{\n \n \n \n \n \n \n \n,\nor \nn\nn\nn\nn\ni\ni\ny\nx\nC\nx\nC\ny\nx\nx\npol\ny\nx\nx\npol\n0\n1\n)\n,\n(\n1\n)\n(\n~\n)}\n,\n('\n/)\n,\n(\"\n{\n2 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n.\nTheorem V :\nLet\n),\n2\n(mod\n0\n)\ndim(\n,\n)\ndim(\n \n \n \nm\nz\nn\nx\n3,2\n,\n1\n)\n,\n(\n \n \n \nh\ny\nx\nC\nn\nh\nh\n \n \n \n. \nThen\n\nn\ns\ns\nm\nz\nz\ny\nx\nC\nDiag\ny\nz\nC\nDiag\ny\nx\nC\ncoper\ny\nz\nC\ny\nz\nC\ny\nx\nC\nx\nz\ngen\n1\n)\n2\n/\n1\n(\n2\n/\n)\ndim(\n3\n3\n)\ndim(\n2\n2\n)\n1\n(\n*\n)\n,\n(\n))\n(\n)\n,\n(\n),\n(\n)\n,\n(\n(\n)1\n(\n)\n1\n)\n,\n(\n,\n1\n)\n,\n(\n,\n)\n)\n,\n(\n(,\n,\n(\n \n \n \n \n \n \n \n \n \n \nProof:\nIt follows from the Binet-Minc identity for \n)\n,\n(\nB\nA\ncoper\nTheorem VI :\n Let\nn\nm\nz\nn\ny\nn\nx\n \n \n \n \n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n. \nThen there exist scalar constants \n3\n2, \n \n such that the system of functions in \n \n,\n, y\nx\n($) \n \n \n \n \n \n \n \n \n \n \n \n3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n3,2\n,\n)\n,\n(\n:\n,..,\n1\n)\n(\nq\ns\nq\ny\nx\nC\nx\nz\nC\ns\ny\nz\nC\nm\nk\ns\nk\nq\ns\nk\n \n \nis algebraically absolutely independent in the region \n)\n( \n \n \n \n \n \n \n \n \n \n3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nn\ns\ns\nn\nn\nn\n \n \n \n \n \nProof:\nlet us consider the Jacobian matrix which is\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n,\n)}\n,\n(\"\n{\n)\n)\n,\n(\n)(\n,\n(\n)\n)\n,\n(\n)(\n,\n(\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n)\n,\n(\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n)\n2\n,1(\n)\n1\n(\n3\n2\n \n \n \n \n \n \ny\nx\ny\nx\nx\npol\ny\nx\nC\nx\nz\nC\ny\nx\nC\nx\nz\nC\ny\nx\nC\nx\nz\nC\ny\nz\nx\nC\ny\nz\nx\nC\nn\ni\n\n)\n2\n(\n2\n2\n2\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n3\n(\n3\n)\n3\n(\n2\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n)\n2\n(\n)\n(\n3\n2\n)\n2\n(\n3\n2\n0\n)\n,\n(\n)\n1\n)\n,\n(\n1\n)\n(\n~\n(\n)\n(\n~\n0\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)\n(\n)\n,\n(\n0\n)\n)\n,\n(\n(\n)\n(\n)\n,\n(\nn\nn\nn\nn\nn\nm\nn\nm\nn\nn\nm\nn\nm\ny\nx\nC\ny\nx\nC\nx\nC\nDiag\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\ny\nx\nC\nDiag\nDiag\ny\nz\nx\nC\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere -s denote some matrices of appropriate dimensions. For the purpose of \nsimplicity, let’s permute in a certain way the Jacobian’s rows and multiply some \nof them by -1, thus reducing it to the form \n \n \n \n \n \n \n \n \n22\n21\n)\n2\n(\n)\n2\n3\n(\n11\n0\nA\nA\nA\nn\nn\nm\n \n where\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n22\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\nA\n \n \n \nIf both the matrices are nonsingular then the Jacobian matrix is nonsingular too. \nIf F is the basic field, let’s consider them over \n)\n( \nF\n for \n)).\ndim(\n(mod\n0\n)\ndim(\n,\nover \n \n vectors\nare\n )\n(\n,\n,\n,\n \nwhere\n,\n,\n1\n,\n,\n1\n,\n]\n[\n]\n[\n]\n1\n[\n0\n]\n[\n)\ndim(\n/)\ndim(\n2\n]\n0\n[\n0\n]\n[\n)\ndim(\n/)\ndim(\n]\n0\n[\n]\n1[\n]\n0\n[\nx\nx\nF\nk\ny\nx\nh\nh\ny\ny\ny\nh\nx\nx\nx\nx\nk\nk\nk\nk\nk\nh\nx\nk\nk\nk\nh\nx\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFirst we’ll show how an -point (satisfying the region’s conditions \n)\n( )\nof such a type is built and then we’ll prove that in the generic case it gives \nnonsingular \n11\nA ,\n22\nA .\nIn order to build such an -point, it’s sufficient that the Jacobian matrix of \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n(\n)\n,\n(\n0\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n1\n)\n,\n(\n1\n)\n(\n~\n(\n)\n(\n~\n)\n(\n)\n,\n(\n)\n)\n,\n(\n(\n3\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n3\n(\n)\n3,2\n(\n2\n2\n2\n2\n2\n3\n2\n3\n2\n11\n \n \n \n \nDiag\ny\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nx\nC\nDiag\nx\nC\nDiag\ny\nx\nC\ny\nx\nC\nDiag\nA\nn\nm\nn\nm\nn\nn\n\nthe system of functions representing the left part of \n)\n( exists and is nonsingular\nat a certain approximation of the power series (in ) for\n \n,y\nwhich satisfies\nthe region’s conditions up to some degree of (then all the other members of \nthe power series are computable via corresponding linear equations involving \nthe Jacobian matrix calculated for the approximation). Let’s consider only the first \ntwo members of the power series for y (i.e. \n]\n0\n[y\n+\n \n]\n1\n[y\n) and the first member of the \none for (i.e. \n]\n0\n[ \n).\nFor \n11\nA , let’s multiply its first two block-rows by \n2\n and then consider its\n0\nlim\n \n \n. In this limit let’s substitute the matrix received via summing up all \nthe columns corresponding to the same \nrh in \n)\ndim(\n,...,\n1\n,\n]\n0\n[\nh\nr\ny\n \n, for its \nfirst block-column (such a transformation can’t enlarge the rank) .\nAnd now \n22\nA :\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n(\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n22\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nC\nDiag\ny\nx\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\nA\n \n \n \n \n \n \n \nP\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nC\ny\nx\nC\nDiag\ny\nx\nC\nP\nI\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\ny\nz\nC\ny\nx\nC\nDiag\ny\nx\nC\nP\nDiag\ny\nx\nC\nn\nm\nn\nm\nn\nm\ni\nj\nj\nn\nn\nm\nn\nn\nn\nn\nn\nm\ni\nj\nj\nn\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)}\n)\n(\n)\n,\n(\n)\n,\n(\n)\n(\n)\n,\n(\ndet(\n)\ndet(\n)1\n(\n{\n0\n0\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)}\n)\n(\n)\n,\n(\n)\n,\n(\n)\n(\n)\n,\n(\ndet(\n)\ndet(\n)1\n(\n{\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n\\\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n\\\n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere \n \n \n \n \n \n \n \n)\n,\n(\n)\n(\n)\n,\n(\ny\nx\nC\nDiag\ny\nx\nC\nP\n \n .\nSince we use the expression \n)\ndet(\n1\nP , first let’s prove the non-singularity of P\nin the generic case. It’s clearly seen for \n)\ndim(\n)\ndim(\nx\nh \n. \nCorollary VI.1 :\n)\n,\n,\n,\n,\n(\n]\n2\n[\n]\n1[\n \n \n \nu\nz\ngen \n (which, according to the Binet-Minc identity, \nis a function in the essential variables: the Cauchy - Binet - Minc weights\n)\n3\n,...,\n,2,1\n(\n)\n,\n(\nq\ns\nq\nu\nz\nC\ns\ni\nq\n \n \n \n \n \n and the star - functional weights\n]\n2\n[\n]\n1\n[ ,\ni\ni\n \n \n)\n\nis computable in polynomial time for any values of the essential (and, hence, \noriginal) variables via a neighbouring computation based on Theorems V, VI \nas the following:\nthe parameterization (of the essential variables) in \n \n,\n, y\nx\n:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n2,1\n,\n1\n)\n,\n(\n)\n2\n(mod\n0\n)\ndim(\n)\n(\n1\n1\n]\n[\nq\ns\nq\ny\nx\nC\nx\nz\nC\nu\nz\nC\ns\ny\nz\nC\nm\nz\ns\ns\nq\nm\ns\ns\ns\n \n \n \n \n \n \nthe region : \n \n \n \n \n \n \n \n \n \n \n \n3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\n)\ndim(\n2\n)\ndim(\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nx\ny\nn\nn\ns\ns\nn\nn\nn\n \n \n \n \n \nHence the final formula for \n)\n,\n,\n,\n,\n(\n]\n2\n[\n]\n1[\n \n \n \nu\nz\ngen \n is:\n)\n}\n)\n,\n(\n{,\n}\n)\n,\n('\n{,\n(\n)\n,\n(\n))\n(\n(\n))\n(\n(\n)\n,\n,\n,\n,\n(\n2\n2\n2\n/\n,\n]\n2\n[\n]\n1[\nn\ni\ni\nn\ni\ni\nm\nn\nz\ny\npol\ny\ny\npol\ny\ndis\nt\nx\npol\nz\nW\nper\nx\nt\nW\nper\nu\nz\ngen\n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere: \n)\n2\n(mod\n0\n)\ndim(\n,\n2\n)\ndim(\n2\n)\ndim(\n,1\n3\n)\ndim(\n \n \n \n \n \n \nm\nz\nn\nx\ny\nn\nt\n ,\n \n)\n,\n(\n)\n,\n('\n \n \n \n \n \n \n \ny\nx\npol\nt\npol\n \n \n ,\n the region : \n \n \n \n \n \n \n \n \n \n3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nn\ns\ns\nn\nn\nn\n \n \n \n \n \nthe parameterization : \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n2,1\n,\n1\n)\n,\n(\n)\n(\n1\n)1\n(\n]\n[\nq\ns\nq\ny\nx\nC\nx\nz\nC\nu\nz\nC\ns\ny\nz\nC\ns\nq\ns\nq\nm\ns\ns\ns\n \n \n \n \n \n \n(\n3\n2, \n \n are scalar constants)\n\nTheorem VII :\n))\n0\n0\n1\n1\n0\n0\n,\n0\n1\n0\n0\n1\n1\n,\n1\n1\n1\n)\n/\n1(\n,\n,\n1\n1\n1\n1\n1\n(\n(\nlim\n)\n,\n,\n,\n1\n1\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n*\n)\ndim(\n)\ndim(\n)\ndim(\n2\n0\n)\ndim(\n)\ndim(\n~\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nw\nw\nv\nv\nu\nu\nw\nw\nv\nv\nu\nu\nv\nv\nu\nw\nv\nv\nu\nu\nw\nv\nu\nv\nu\nt\nv\nv\nu\nh\nh\nv\nv\nu\nu\ngen\nt\nh\nv\nu\ngen\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nTheorem VIII :\n))\n,\n,\n,\n1\n1\n,\n1\n(\n(\nlim\n)\n,\n,\n,\n,\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n0\n \n \n \n \n \n \nt\nh\nw\nw\nz\ngen\nt\nh\nw\nz\ngen\nu\nu\nu\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nTheorem IX :\n)\n(\n))\n,\n,\n,\n,\n,\n(\n(\nlim\n)\n(\n)\n,\n,\n,\n0\n,\n(\n)\ndim(\n)\ndim(\n2\n0\n)\ndim(\n1\n)\ndim(\n~\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \ny\nt\nx\nh\nw\nz\ngen\ncoef\nt\nh\nw\nz\ngen\nh\nw\nx\nj\nj\nu\n \nwhere is a formal scalar variable, and for \n3,2,1\n,2,1\n \n \nq\ns\n:\n(i) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\ndim(\n)\ndim(\n)\ndim(\n3\n1\n0\n)1\n(\n)\n2\n(\n)\n(\n)1\n(\n)\n,\n(\n)\n,\n(\n)\n(\nz\nt\nw\ns\ns\nq\ns\nq\ns\nq\ns\nx\nt\nz\nC\nx\nw\nC\n \n \n \n \n \n \n \n \n(ii) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n1\n(\n)\ndim(\n)\ndim(\n)\ndim(\n3\n)1\n(\n0\n)1\n(\n)\n2\n(\n)\n(\n)\n,\n(\n)\n,\n(\n)\n,\n(\n)1\n(\n)\n(\n \n \n \n \n \n \n \n \nq\nq\ns\nq\ns\ny\nx\nC\ny\nh\nz\nC\ny\nw\nC\nz\nh\nw\ns\nq\ns\ns\n \nProof:\nAccording to the definition, \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n))\n,\n,\n,\n,\n,\n(\n(\n)\ndim(\n \n \n \n \ny\nt\nx\nh\nw\nz\ngen\ncoef\nh\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nJ\nJ\nw\nI\nJ\nI\nJ\nI\ny\nK\nx\nh\nJ\nI\nK\nh\nK\nK\nw\nz\nesum\nw\nz\ny\nt\nx\nh\nesum\n)\n)\n0,0\n(\n,\n(\n)\n)\n0,0\n(\n)\n0,\n(\n)\n,\n(\n)\n0,\n(\n)\n,0\n(\n)\n,0\n(\n,\n(\n|\n|\n)\ndim(\n\\\n\\\n\\\n|\n|\n|\n|\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n|\n|,\n\nand in such a case \n)\n)\n0,\n(\n)\n,\n(,\n(\n)\ndim(\n \n \n \n \n \ny\ny\n is a right prolongation. Then, according to the\nprolongation-derivative principle and the equality (ii), the first multiplier of the expression \nunder the summation signs is \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n)\n0,\n(\n)\n0,0\n,\n(\n)\n0,0\n,0\n(\n)1,\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n|\n|\n3\n2\n1\n1\n|\n|\n)\ndim(\n)\n1\n(\n)\ndim(\n)\n(\nJ\nI\nK\nx\nh\nJ\nI\nK\nw\nz\nt\nx\nh\nesum\n \n \n \n \n \n \n)\n)\n0,\n(\n)\n0,0\n,\n(\n)\n0,0\n,0\n(\n)1,\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n1\n|\n|\n3\n2\n1\n1\n|\n|\n)\ndim(\n)\ndim(\n)\ndim(\n1\n)\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nI\nK\nx\nh\nJ\nI\nK\nx\nj\nj\nw\nz\nt\nx\nh\nesum\n \n \n \n \n \n \n \nThen \n)\n)\n,0\n(\n)\n,\n(,\n(\n)\ndim(\n \n \n \n \n \nx\nx\n is a left prolongation, and we can apply the \nprolongation-derivative principle again, this time together with the equality (i), \nhence receiving\n \n \n)\ndim(\n1\n1\nx\nj\nj\n \n)\n)\n0,0\n,\n(\n)\n0,0\n(\n)\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n|\n|\n2\n2\n1\n|\n|\n)\ndim(\n)\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nI\nK\nh\nJ\nI\nK\no\nw\nz\nt\nh\nesum\n \n \n \n \nBecause the second multiplier doesn’t depend on , we can apply the operator \n...)\n(\nlim\n)\ndim(\n2\n0\nw\n \n \nto the first one only, hence receiving\n))\ndim(\n|\n(|\n)\n)\n0,0\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n))\ndim(\n|\n(|\n)\n)\n0,0\n,\n(\n)\n0,0\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n)\ndim(\n1\n)\ndim(\n|\n|\n)\ndim(\n)\ndim(\n1\n)\ndim(\n|\n|\n)\ndim(\n)\n(\nw\nJ\nw\nz\nt\nh\nesum\nw\nJ\nw\nz\nt\nh\nesum\nx\nj\nj\nw\nI\nK\nh\nI\nK\nx\nj\nj\nw\nI\nK\nh\nI\nK\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere, taking into account that the third sum\n \n \n)\ndim(\n|\n|,\nh\nK\nK\nruns over sets of cardinality \n)\ndim(h (thus “balancing” the patterns \n)\ndim(\n)\n,0\n(\nh\n \n and \nK\n \n)\n0,\n( \n), the last passage \nis due to the requirement (following from the E-sum’s definition) that the total\n\nbalance is to be 0. So, after subjecting the product of the two multipliers to the \noverall summation \n \n \nI\nJ\nh\nK\nK\n)\ndim(\n|\n|,\n, we get the theorem’s claim. \nTheorem X (for any characteristic) :\n1)\n \n \n \n \n \n \n \nP\nj\ni\nj\ni\nz\nPart\nP\nz\nz\nz\nC\nper\n}\n,\n{\n2\n)})\ndim(\n,...,\n1\n({\n)\n(\n1\n))\n(\n~\n(\n2\n ,\nwhere for a set N\n)\n(\n2 N\nPart\nis the set of its perfect matchings. \n2) Let \n)\ndim(\n)\ndim(\ny\nx \n. Then\n)\n(\n)\n(\n)\ndim(\n1\n)\n,\n(\n)\n,\n(\n)\n(\n~\n))\n,\n(\n(\n))\n,\n,\n(\n~\n(\n)\n(\nz\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nz\ny\nx\nC\nper\n \n \n \n \n \nProof:\n1) first let’s prove that, if for a \nd\nd \n-matrix A we define \n \n \n \n \n \nd\nH\nd\ni\ni\ni\ndef\na\nA\nham\n \n \n1\n)\n(\n,\n)\n(\nwhere \nd\nH is the set of d -permutations with one cycle, \n0\n))\n(\n~\n(\n \nu\nC\nham\nwhen \n2\n)\ndim(\n \nu\n.\nSuppose \n2\n)\ndim(\n \nu\n. Let’s partition the set \nd\nH on disjoint subsets consisting of d -1 \nd -cycles which differ from each other only in the position of the element d . Each of those \nsubsets can be received via taking a ( d -1)-cycle \n)\n,...,\n(\n1\n1\n \n \ndh\nh\nh\n with elements from the set \n}\n1\n,...,\n1\n{\n \nd\n and alternate placing d between neighbors in h (altogether there are d -1 options).\nIn such a case the sum of the corresponding d -1 summands in the expression \n))\n(\n~\n(\nu\nC\nham\n will be\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n1\n1\n1\n1\n1\n1\n1\n1\n0\n)\n1\n1\n(\n)\n1\n(\n1\n)\n)(\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\nd\ni\nd\nk\nd\nh\nd\nh\nd\ni\nh\nh\nh\nh\nd\nk\nh\nd\nd\nh\nh\nh\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nk\nnext\nk\ni\nnext\ni\ni\nnext\ni\nk\nnext\nk\nk\nnext\nk\nwhere \n \n \n \n \n \n \n \n \n1\n,1\n1\n,1\n)\n(\nd\nt\nd\nt\nt\nt\nnext\n . \nTaking into account that for an \nn\nn \n-matrix A \n \n \n \n \n \n})\n,...,\n1\n({\n)\n,\n(\n)\n(\n)\n(\nn\nPart\nP\nP\nI\nI\nI\nA\nham\nA\nper\n \nand \n)\n(\n~\n))\n(\n~\n(\n)\n,\n(\nI\nI\nI\nz\nC\nz\nC\n \n, we get the first claim of the theorem;\n2) we apply the induction on \n)\ndim(z . \nFirst let’s prove the induction’s basis for \n1\n)\ndim(\n \nz\n. Let z be scalar. Then\n\n)\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n1\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n(\nlim\n)))\n,\n(\n(\n1\n))\n,\n(\n(\n(\nlim\n))\n,\n,\n(\n~\n(\n2\n2\n0\n0\ny\nx\nC\ny\nx\nC\nz\ny\nz\nx\nC\nz\ny\nz\nx\nC\ny\nx\nC\nper\nz\ny\nz\nx\nC\nper\nz\ny\nx\nC\nper\n \n \n \n \n \n \n \n)\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n1\n))\n(\n1\n))\n,\n(\n)\n,\n(\n(\n1\n))(\n,\n(\ndet(\n1\n)1(\n))\n,\n(\ndet(\n1\n(\nlim\n2\n2\n)\ndim(\n2\n2\n0\ny\nx\nC\ny\nx\nC\nO\ny\nz\nC\nx\nz\nC\ny\nx\nC\nO\ny\nx\nC\nx\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(the second passage is due to the Borchardt formula).\nAnd now the induction’s step:\nlet \n1\n,\n)\ndim(\n \n \nm\nz\nm\nz\nis a scalar. Then\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\n1\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\nlim\n)\n))\n,\n,\n(\n~\n(\n1\n))\n,\n,\n(\n~\n(\n(\nlim\n))\n,\n,\n(\n~\n(\n)\ndim(\n)\ndim(\n1\n1\n1\n1\n0\n1\n1\n0\n1\nz\nz\nm\nm\nm\nm\nm\nm\nm\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nz\ny\nz\nC\nz\nx\nz\nC\nDiag\nz\nC\nper\nz\ny\nz\nx\nC\nper\nz\ny\nx\nC\nper\nz\nz\ny\nz\nx\nC\nper\nz\nz\ny\nx\nC\nper\n \n \n \n \n \n \n \n \n \n)\n(\n(\n)\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\n1\n))\n(\n))\n0\n1\n))\n,\n(\n)\n,\n(\n(\n(\n)\n(\n~\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n(\n1\n))\n,\n(\n)\n,\n(\n(\n1\n))(\n,\n(\n(\n1\nlim\n)\ndim(\n2\n)\ndim(\n1\n)\ndim(\n2\n)\ndim(\n1\n1\n0\nz\nz\nm\nz\nx\nm\nm\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nO\ny\nz\nC\nx\nz\nC\nDiag\nz\nz\nC\nper\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\nO\ny\nz\nC\nx\nz\nC\ny\nx\nC\nper\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThe last passage is due to the part (1) of this theorem.\nTheorem XI :\n))\n,\n,\n,\n1\n1\n0\n,\n1\n1\n1\n1\n1\n1\n(\n(\nlim\n)\n,\n,\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n)\ndim(\n2\n0\n^\n \n \n \n \n \n \nt\nh\nu\nu\nu\ngen\nt\nh\nu\ngen\nu\nu\nu\nu\nu\nu\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nProof:\n\n)\nX(2)\nTheorem\nuse\n(we\n))\n,\n,\n(\n~\n(\n))\n(\n~\n(\n)\n1\n(\n)\n,\n,\n,\n1\n1\n,\n(\n))\n,\n,\n,\n1\n1\n0\n,\n1\n1\n1\n1\n1\n1\n(\n(\nlim\n)\ndim(\n|\n|,\n\\\n|\n|\n)\ndim(\n~\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n)\ndim(\n2\n0\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nJ\nj\nJ\nh\nJ\nJ\nI\nJ\nI\nI\nu\nu\nu\nu\nu\nu\nu\nu\nt\nh\nC\nper\nu\nC\nper\nt\nh\nu\ngen\nt\nh\nu\nu\nu\ngen\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nj\nJ\nJ\nh\nJ\nJ\nu\nJ\nI\nJ\nj\nJ\nh\nJ\nJ\nJ\nu\nJ\nI\nI\nI\nI\nI\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nDiag\nper\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nDiag\nu\nC\nper\nu\nC\nper\n \n \n))\n,\n(\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n(\n)\nX(1)\nTheorem\nuse\nwe\n(\n))\n,\n(\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n(\n~\n(\n)\n1\n(\n)\ndim(\n|\n|,\n)\ndim(\n)\ndim(\n|\n|,\n)\ndim(\n\\\n\\\n\\\n|\n|\n \n \n)\n,\n,\n,\n(\n))\n,\n(\n(\n))\n1\n)\n,\n(\n1\n)\n,\n(\n(\n(\n^\n)\ndim(\n1\n)\ndim(\n)\ndim(\n)\ndim(\n|\n|,\n \n \nt\nh\nu\ngen\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nu\ni\nJ\nj\nJ\nJ\nu\nJ\ni\nu\ni\nh\nJ\nJ\n \n \n \n \n \n \n \n \n \n \n \n \nTheorem XII :\nLet \n.\n0\n)\n(\n)\n,\n(\n),\ndim(\n)\ndim(\n)\ndim(\n2\n)\n1,\n3\n1\n(\nh\nDiag\nt\nh\nC\nh\nu\n \n \n \n \n \n Then\n)\n1\n,\n,\n,\n1\n(\n)\n1\n1(\n))\n,\n(\n)\n(\n)\n,\n(\n(\n3\n^\n)\ndim(\n3\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nt\nt\nh\nu\ngen\nh\nt\nC\nDiag\nt\nu\nC\nper\nm\nm\nh\n \nProof:\nIt follows from the generalized Binet-Minc identity for \n)\n(\nˆ\nA\nr\ne\np\nB\n, due to the fact \nthat \n0\n)\n1\n(\n)\n,\n(\n,3\n|\n|,\n)\n(\n)})\ndim(\n2\n,...,\n1\n{,\n()\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nt\nI\nDiag\nt\nt\nh\nC\nscal\nI\nI\n \n \n(by the construction\nand the condition \n)\ndim(\n2\n)\n1,\n3\n1\n(\n0\n)\n(\n)\n,\n(\nh\nDiag\nt\nh\nC\n \n \n \n \n), while \n)\ndim(\n)\ndim(\nh\nu \n.\nCorollary XII.1 :\nLet \n,0\n)\n)\n,\n(\n)\n,\n(\ndet(\n),\n2\n(\n)\ndim(\n,\n3\n)\ndim(\n)\ndim(\n)\n,...,\n1(\n)\n1,\n3\n1\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nt\nu\nC\nt\nh\nC\nn\nn\nt\nn\nh\nu\nn\nm\nM is an \nn\nn -matrix. Then\n)\n1\n1\n)\n...\n0\n)\n,\n(\n))\n,\n(\n(\n.........\n)\n,\n(\n))\n,\n(\n(\n)\n,\n(\n(,\n,\n,\n1\n(\n)\n1\n1(\n)\n(\n)\n},\n,...,\n1\n({\n)\n1\n},\n,...,\n1\n({\n2\n1\n)\n1\n,...,\n(\n)\n,...,\n1\n(\n1\n)\n1\n,...,\n(\n1\n)\n,...,\n1(\n)1,\n3\n1\n(\n3\n^\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nn\nn\nn\nn\nn\nn\nT\nn\nn\nn\nT\nn\nn\nM\nM\nt\nu\nC\nh\nu\nC\nt\nu\nC\nh\nu\nC\nt\nh\nC\nt\nt\nh\nu\ngen\nM\nper\nm\n\nReferences:\n1) L.G. Valiant. The Complexity of Computing the Permanent.\nTheoret. Comp. Sci. 8, 189-201, 1979.\n2) Okada Soichi. Generalizations of Cauchy’s Determinant Identity \nand Schur’s Pfaffian Identity.\nhttp://citeseer.ist.psu.edu/cache/papers/cs2/125/http:zSzzSzwww.math.kobe-\nu.ac.jpzSzpublicationszSzrlm18zSz9.pdf/generalizations-of-cauchy-s.pdf\n3) Richard A. Brualdi, Herbert J. Ryser, Combinatorial Matrix Theory, \nCambridge University Press, Cambridge, 1991. Comm. Computer Algebra \n39 (2005), pp. 61–64.\n 4) Henryk Minc, Permanents, Addison-Wesley, Reading, 1978.\n 5) http://en.wikipedia.org/wiki/Pfaffian\n 6) http://en.wikipedia.org/wiki/Cauchy_determinant\n 7) Mark Jerrum. The computational complexity of counting. \nThe International Congress of Mathematicians, Zurich, August 1994.\nhttp://www.lfcs.inf.ed.ac.uk/reports/94/ECS-LFCS-94-296/index.html","paragraphs":[{"paragraph_id":"p1","order":1,"text":"Vadim Tarin\n A Polynomial-time Algorithm for \nComputing the Permanent in GF( q\n3 ) \nAbstract: \nA polynomial-time algorithm for computing the permanent in any field of characteristic 3 is \npresented in this article. The principal objects utilized for that purpose are the Cauchy and \nVandermonde matrices, the discriminant function and their generalizations of various types. \nClassic theorems on the permanent such as the Binet-Minc identity and Borchadt's formula \nare widely applied, while a special new technique involving the notion of limit re-defined for \nfields of finite characteristics and corresponding computational methods was developed in \norder to deal with a number of polynomial-time reductions. All the constructions preserve a \nstrictly algebraic nature ignoring the structure of the basic field, while applying its infinite \nextensions for calculating limits.\nA natural corollary of the polynomial-time computability of the permanent in a field of a \ncharacteristic different from 2 is the non-uniform equality between the complexity classes \nP and NP what is equivalent to RP=NP( Ref. [1]).\n Unless specified otherwise, \n all the results are for fields of Characteristic 3.\nDefinitions and denotations.\nFor \nm\nn \n-matrices \nm\nn\nh\nij\ndef\nh\nm\nn\nij\nij\ndef\na\nA\nb\na\nB\nA\nB\nA"},{"paragraph_id":"p2","order":2,"text":"}\n{\n,\n}\n{\n,\n,"},{"paragraph_id":"p3","order":3,"text":","},{"paragraph_id":"p4","order":4,"text":"q\nq\nh\nh\ndef\nh\nh\nA\nA\nA\n...\n1\n1\n)\n,...,\n(\n (for real numbers \nqh\nh\nh\n,...,\n,\n1\n)\n(the Hadamard product, degree and vector-degree),\nfor \n)\n,\n(\n}\n,...,\n1\n{\n},\n,...,\n1\n{\nJ\nI\nA\nm\nJ\nn\nI"},{"paragraph_id":"p5","order":5,"text":"is the sub-matrix of A whose \nset of rows is I and of columns J ,"},{"paragraph_id":"p6","order":6,"text":"for matrices \n]\n[\n]\n1[ ,...,\ns\nA\nA\n with the same number of rows \ns\nr\nr\nA\n1\n]\n["},{"paragraph_id":"p7","order":7,"text":"denotes \nthe matrix"},{"paragraph_id":"p8","order":8,"text":"]\n[\n]\n1\n[\n|\n...\n|\ns\nA\nA\n,"},{"paragraph_id":"p9","order":9,"text":"n\ni\nij\nm\nj\na\nA\nscal\n1\n1\n)\n(\n ,"},{"paragraph_id":"p10","order":10,"text":";\ns)\nfor vector\n(as\n)\n('\n),\n(\nsimply \n write\n will\nwe\n1\n)\ndim(\nfor\n,)\n(\n...\n)\n('\n,)\nmatrix\ne\nVandermond\nthe\n(\n)\n(\n)\n(\n,\n)\n(\n]\n[\n]\n[\n]\n[\n)\ndim(\n1\n]\n[\n)]\n[dim(\n)\n1\n,...,\n0\n(\n]\n[\nt\nvan\nt\nvan\nt\nt\nVan\nt\nd\nd\nt\nd\nd\nt\nVan\nt\nVan\nt\nVan\nt\nt\nVan\nh\nh\nh\nt\ndef\nh\nt\ndef\nh\nT\ndef\nh"},{"paragraph_id":"p11","order":11,"text":")\n2\n(mod\n1\n)\ndim(\n,\n)\n(\n)\n(\n),\n2\n(mod\n0\n)\ndim(\n,\n)\n(\n)\n(\n)\n(\n]\n2\n/)\n1\n)\n[(dim(\n]\n2\n/)1\n)\n[(dim(\n]\n2\n/)\n[dim(\n]\n2\n/)\n[dim(\nt\nt\nVan\nt\nVan\nt\nt\nVan\nt\nVan\nt\nW\nt\nt\nt\nt\ndef\n,\n)\n(\n)\n,\n(\n)\ndim(\n1\n)\ndim(\n1"},{"paragraph_id":"p12","order":12,"text":"u\ni\nv\nj\nj\ni\ndef\nv\nu\nv\nu\npol\nfor a scalar \n)\n,\n(\n)\n,\n(\"\n,)\n,\n(\n)\n,\n('\n,\n2\n2\nv\npol\nd\nd\nv\npol\nv\npol\nd\nd\nv\npol\ndef\ndef"},{"paragraph_id":"p13","order":13,"text":";\nfor an appropriate set I , \nIu\nis the sub-vector of u consisting of the coordinates \nwith indexes from I (including the case when I is a multi-set, with correspondingly \nduplicated coordinates),\nI\nu\\\nis its compliment in u ;\nfor a scalar , \nm\nn"},{"paragraph_id":"p14","order":14,"text":"is the \nm\nn \n-matrix all whose entries equal ,\nfor \n1"},{"paragraph_id":"p15","order":15,"text":"m\nwe will write simply \nn\n ;`\nfor a set H , \n)\n(H\nPart\n is the set of its partitions on nonempty sets;\nfor a finite sequence d, |d| denotes the number of its members;\nfor a real number"},{"paragraph_id":"p16","order":16,"text":"0\n,1\n0\n,0\n)\n(\n,\nu\nu\nu\nu \n ;\nwhere \n)\ndim(\n|\n|,\n1,\n|\n|,\n1,\n)\ndim(\n)\ndim(\n)}\n,...,\n,\n,...,\n{(\n)}\n,\n{(\n}\n{\na\ni\ni\ni\ni\na\ni\ni\na\ni\ni\ni"},{"paragraph_id":"p17","order":17,"text":"is a \nmatrix)\nextension \nCauchy \nthe\n(\n}\n}\n1\n)!\n)(\n!\n(\n1\n)){\n(\n1\n{(\n)\n,\n(\n);\nmatrix\nCauchy \nthe\n(\n}\n1\n{\n)\n,\n(\n)\ndim(\n)\ndim(\n|\n||\n|\n,\n,\n)\ndim(\n)\ndim(\n,\n,\n,\n,\na\na\nj\ni\nj\ni\nl\nj\nk\ni\ndef\ny\nx\nj\ni\ndef\nj\ni\nl\nj\nl\nj\nk\ni\nk\ni\na\na\na\nd\nd\na\nd\nd\nj\ni\na\nE\ny\nx\ny\nx\nC"},{"paragraph_id":"p18","order":18,"text":")\ndim(a -vector whose coordinates are pairs of non-decreasing nonnegative integer \nfinite sequences, including the empty one. The rows and columns containing the\nterm \nia will be called the \nia -extension (and \nia itself the extension’s value );\n)\n,\n(\ni\ni\ni"},{"paragraph_id":"p19","order":19,"text":"we’ll call the extension-degree of \nia , where \ni is the left \nextension-degree, \ni\n is the right one, \n|\n|\ni \n is the extension’s height , \n|\n|\ni"},{"paragraph_id":"p20","order":20,"text":"is its width , and \n|\n|\n|\n|\n)\n(\ni\ni\ndef\ni\nbal"},{"paragraph_id":"p21","order":21,"text":"is its balance . If \n0\n|\n|\n|\n|"},{"paragraph_id":"p22","order":22,"text":"i\ni"},{"paragraph_id":"p23","order":23,"text":"then we’ll say that the matrix has an \nia -singularity. Singularities with \n(0,0)"},{"paragraph_id":"p24","order":24,"text":"i \nwill be called simple. The sum"},{"paragraph_id":"p25","order":25,"text":")\ndim(\n1\n)\n(\n)\n(\na\ni\ni\nbal\nbal"},{"paragraph_id":"p26","order":26,"text":"will be called the total balance\nof the extension-degrees’ vector ."},{"paragraph_id":"p27","order":27,"text":"field.\nbasic"},{"paragraph_id":"p28","order":28,"text":"over the\nfunction \n-\nweight\nits\nis\n )\n(\ndegrees,\n-\nextension\nof\nset \na\nis"},{"paragraph_id":"p29","order":29,"text":"),\ndim(\n1,...,\nfor \nwhere,\n))\n,\n(\n(\n))\n(\n(\n))\n(\n(\n..."},{"paragraph_id":"p30","order":30,"text":"expression\n the\nunderstand\nll\n we'\nsum\n-\nE\nan \nBy \n1\n1\n)\ndim(\n)\ndim(\n)\ndim(\n1\ni\ni\ni\na\ni\ni\ni\nweight\na\ni\na\nE\nper\nbal\nweight\na\na"},{"paragraph_id":"p31","order":31,"text":"The formal sum"},{"paragraph_id":"p32","order":32,"text":"i\ni\ni\ni\ni\ni\nweight"},{"paragraph_id":"p33","order":33,"text":")\n(\n will be defined as the extension- plane\nof \nia , \n)\n(\ni\ni\nweight \n will also be called the \nweight\nplane -\n of \ni in \ni , the pair \n)\n,\n(\ni\nia \nwill be called an extension-variety and thus we can also talk about the E-sum of \na on the extension-planes \n),\ndim(\n1,...,"},{"paragraph_id":"p34","order":34,"text":",\na\ni\ni"},{"paragraph_id":"p35","order":35,"text":"while the pair of vectors \n)\n,\n("},{"paragraph_id":"p36","order":36,"text":"a\nwill be called a vector-variety. If \ni has the form \n)\n0,\n(\n)\n,\n("},{"paragraph_id":"p37","order":37,"text":"i \n then we’ll say \nthat the E-sum is right-prolonged on \nia , if \n)\n,0\n(\n)\n,\n("},{"paragraph_id":"p38","order":38,"text":"i \n then left-prolonged,\nand the corresponding patterns of \n)\n,\n("},{"paragraph_id":"p39","order":39,"text":"a\n (i. e. sub-vectors of the vector-variety) will be \ncalled right and left prolongations (correspondingly).\nWe will denote the E-sum of a on by \n)\n,\n("},{"paragraph_id":"p40","order":40,"text":"a\nesum\n (we’ll use the brackets"},{"paragraph_id":"p41","order":41,"text":"only \nfor left and right extension-degrees whose cardinalities exceed 1);\n)\n)\n0,0\n(,\n(\n)\n(\n~\n),\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n)\n,\n,\n(\n~\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\nz\ndef\nz\ny\nx\ndef\nz\nE\nz\nC\nz\ny\nx\nE\nz\ny\nx\nC"},{"paragraph_id":"p42","order":42,"text":";\nfor an \nr\nn -matrix A and an\nr\nm -matrix B ,\nm\nn\nm\n,\n2"},{"paragraph_id":"p43","order":43,"text":"is even :\n(the copermanent of A with B ) ,\n)\n(\n)\n(\n)\n,\n(\n})\n,\n{\n},\n,...,\n1\n({\n)\n},\n,...,\n1\n({\n2\n/\n|\n|,\n|\n|,\n:\n,\nJ\nJ\nm\nI\nn\nm\nJ\nn\nI\nJ\nI\nJ\nI\ndef\nB\nper\nA\nper\nB\nA\ncoper"},{"paragraph_id":"p44","order":44,"text":",\n)\n(\n)\n(\n,\n)\ndet(\n)\ndet(\n|\n|,\n)\n},\n,...,\n1\n({\n|\n|,\n)\n},\n,...,\n1\n({"},{"paragraph_id":"p45","order":45,"text":"n\nJ\nJ\nJ\nn\nn\nJ\nJ\nJ\nn\nA\nper\nA\nper\nA\nA"},{"paragraph_id":"p46","order":46,"text":"m\nk\nJ\nj\nkj\nJ\nn\nn\nJ\nJ\ndef\nB\nb\nA\nper\nA\nr\ne\np\n1\n)\n},\n,...,\n1\n({\n|\n|,\n)\n(\n)\n(\nˆ\n(the permanent of A on the base B , or the base-permanent of A on B );\nfor r-vectors"},{"paragraph_id":"p47","order":47,"text":",\n,u\n, nonnegative integers \nq\np \n:"},{"paragraph_id":"p48","order":48,"text":"I\ni\nJ\nj\nj\ni\nJ\nI\nq\nJ\np\nI\nJ\nI\nJ\nI\ndef\nq\np\nu\nVan\nu\nVan\nu\ndis\n)\n)(\n))(\n(\n(\ndet\n))\n(\n(\ndet\n)\n,\n,\n(\n2\n2\n|\n|,\n|\n|,\n:\n,\n,"},{"paragraph_id":"p49","order":49,"text":"(the \nq\np, -discriminantal of u on"},{"paragraph_id":"p50","order":50,"text":",\n) ;\nthe E-generated functions (for vectors with appropriate dimensions):\nthe star-function:"},{"paragraph_id":"p51","order":51,"text":"2\n/)\ndim(\n|\n|,\n,\n,\n}\n,\n,\n{\n\\\n]\n2\n[\n]\n1\n[\n]\n2\n[\n]\n1\n[\n))\n(\n)\n,\n(\n(\n))\n,\n(\n(\n)\n)(\n(\n)\n,\n,\n,\n,\n(\nz\nJ\nI\nJ\nI\nJ\nI\nI\ni\nJ\nJ\nI\nJ\nI\nJ\nI\nJ\nj\nj\ni\ndef\ndef\nDiag\nu\nz\nC\nper\nz\nz\nC\nper\nu\nz\ngen"},{"paragraph_id":"p52","order":52,"text":"the 2-waves-function:\n)\n)\n0,0\n(\n,\n(\n)\n)\n0,0\n(\n)\n0,\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n)\n,\n,\n,\n,\n,\n(\n|\n|\n)\ndim(\n\\\n\\\n\\\n|\n|\n|\n|\n)\ndim(\n)\ndim("},{"paragraph_id":"p53","order":53,"text":"J\nw\nI\nJ\nI\nI\nJ\nJ\nI\ny\nx\nJ\nI\ndef\ndef\nw\nz\nesum\nw\nz\nt\nh\nesum\nt\nh\nw\nz\ngen"},{"paragraph_id":"p54","order":54,"text":")\n0,1(\n)1,0\n(\n)\n0,0\n,\n0,0\n("},{"paragraph_id":"p55","order":55,"text":"where"},{"paragraph_id":"p56","order":56,"text":"def\n.\nWe’ll call the extension-plane (0,0) the wave and the biwave; together with \ntheir values (say \niz and \nj\nw ) such extension-varieties will be called a \niz -wave \nand a \nj\nw -biwave (correspondingly) ; \nthe wave-function (a partial case of the 2-waves-function):\n;\n))\n(\n~\n(\n)\n))(\n,\n,\n(\n~\n(\n)\n,\n,\n,\n,\n,\n(\n)\n,\n,\n,\n,\n(\n)\ndim(\n|\n|,\n\\\n~"},{"paragraph_id":"p57","order":57,"text":"I\nh\nJ\nJ\nJ\nj\nj\nI\nI\ni\ni\nI\nJ\ndef\nz\nC\nper\nz\nt\nh\nC\nper\nt\nh\nz\ngen\nt\nh\nz\ngen"},{"paragraph_id":"p58","order":58,"text":"For both the wave-function and the 2-waves-function \ni\n will be called the i -th ebb, \nand the ebb vector.\nthe base-function:"},{"paragraph_id":"p59","order":59,"text":"I\nt\nu\nC\nI\ni\nh\ni\nh\nh\nt\nh\nh\nt\nu\nC\nh\nu\nC\ndef\nDiag\nt\nh\nC\nr\ne\np\nh\nu\nC\nDiag\nt\nh\nC\nI\nr\ne\np\nt\nh\nu\ngen\nI\n))\n(\n)\n,\n(\n(\nˆ\n))\n1\n)\n,\n(\n(\n(\n)\n)\n(\n)\n,\n(\n0\n0\n(\nˆ\n)\n,\n,\n,\n(\n)\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\n,\n(\n)\n,\n(\n^\n\\"},{"paragraph_id":"p60","order":60,"text":"if F is a basic field and is a formal variable then for"},{"paragraph_id":"p61","order":61,"text":"0\n)\n(\n,\nexist\nt \ndoesn'\n0\n)\n(\n,\n)\n(\nlim\n,\n)\n(\n:)\n(\n]\n0\n[\n0\n]\n[\ng\norder\ng\norder\ng\ng\nq\ng\norder\nF\ng\ng\nq\nk\nk\nk"},{"paragraph_id":"p62","order":62,"text":"By \n)\n),\n(\n(\nx\nx\nf"},{"paragraph_id":"p63","order":63,"text":"we’ll denote the Jacobian matrix \n)\ndim(\n)\ndim(\n}\n)\n(\n{\nx\nf\nj\ni\nx\nx\nf"},{"paragraph_id":"p64","order":64,"text":".\nPreliminaries.\nThe Binet-Minc identity (for any characteristic, Ref. [3], [4]):\nLet A be an"},{"paragraph_id":"p65","order":65,"text":"m\nn\nmatrix, then\n)\n(\n})\n,...,\n1\n({\n1\n)!\n1\n|\n(|\n)1\n(\n)\n("},{"paragraph_id":"p66","order":66,"text":"n\nPart\nI\nm\nj\nI\ni\nij\nn\na\nI\nA\nper\n,\nThe Binet-Minc identity for Characteristic 3:\nLet A be an"},{"paragraph_id":"p67","order":67,"text":"r\nn\nmatrix, then"},{"paragraph_id":"p68","order":68,"text":"})\n,...,\n1\n({\n)\n,\n(\n3\n,\n2\n,1\n)\n(\n)\n(\n)!\n1\n|\n(|\n)1\n(\n)\n(\nn\nPart\nI\nR\nI\nn\nA\nscal\nI\nA\nper\n ,\nwhere:\n)\n(\n3,2\n,1\nH\nPart\n is the set of partitions of the set H on nonempty sets of cardinalities \nnot bigger than 3, \n}\n,...,\n1\n{\nr\nR \n.\nFor our purposes we’ll also use this identity in the form"},{"paragraph_id":"p69","order":69,"text":"})\n,...,\n1\n({\n)\n,\n(\n|\n|\n1\n3\n,\n2\n,1\n)\n(\n)\n(\n)!\n1\n|\n(|\n)1\n(\n)\n(\nn\nPart\nI\nR\nI\nI\nA\nscal\nI\nA\nper\nThe generalized Binet-Minc identity for \n)\n,\n(\nB\nA\ncoper\n :\nLet A , B be an \nr\nn - and \nr\nm -matrix, \n)\n2\n(mod\n0"},{"paragraph_id":"p70","order":70,"text":"m\n, then"},{"paragraph_id":"p71","order":71,"text":"I\nR\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nK\nn\nN\nK\nm\nA\nscal\nI\nB\nA\nscal\nB\nA\ncoper\n)\n(\n)\n(\n)\n(\n)\n)!\n1\n|\n(|\n(\n)\n(\n)1\n(\n)1\n(\n)\n,\n(\n)\n,\n(\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|\n|\n2\n/\n3\n,\n2\n,1\n2\n;\n2\n,1"},{"paragraph_id":"p72","order":72,"text":"where:\n},\n,...,\n1\n{\n},\n,...,\n1\n{\n},\n,...,\n1\n{\nr\nR\nm\nM\nn\nN"},{"paragraph_id":"p73","order":73,"text":"for two disjoint sets \n'\nH and \n\"\nH ,\n)\"\n,'\n(\n2\n;\n2\n,1\nH\nH\nPart\n is the set of partitions of the set \n\"\n' H\nH \non sets having 1 or 2 elements in \n'\nH and 2 elements in \n\"\nH (those sets are denoted here as \npairs \n\"\n\"\n,'\n'\n),\n\"\n,'\n(\nH\nI\nH\nI\nI\nI"},{"paragraph_id":"p74","order":74,"text":").\nProof:\nthe right part can be re-written as \n)\n(\n)\n(\n)\n(\n)1\n(\n)!\n1\n|\n(|\n)\n(\n)\n,\n(\n1\n|\n|\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n3\n,\n2\n,1\n2\n;\n2\n,1"},{"paragraph_id":"p75","order":75,"text":"I\nR\nI\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nN\nK\nA\nscal\nI\nB\nA\nscal\n. \nLet’s denote it by \n)\n,\n(\nB\nA"},{"paragraph_id":"p76","order":76,"text":".\nThen, by the construction,\n)\n(\n}\n,...,\n{\n)\n(\n}\n,...,\n{\n}\n,...,\n1\n{\n,\n,\n'\n\"\n\"\n'\n|\n|\n|\n|\n\\\n\"\n|\n|\n\"\n1\n'\n|\n|\n'\n1\n\"\n1\n'\n1\n\"\n|\n|\n'\n|\n|\n'\n\"\n\"\n'\n)\n1,\n1\n(\n...\n)\n,\n(\nM\nPart\nI\nI\nN\nPart\nI\nI\nr\nJ\nI\nI\nI\nI\nJ\nj\nI\ni\nI\ni\nj\ni\nj\ni\nI\nI\nJ\nJ\nJ\nJ\nj\nj\nj\nj\nb\na\nB\nA"},{"paragraph_id":"p77","order":77,"text":"where"},{"paragraph_id":"p78","order":78,"text":"\\}\n{\n means the set of sets with removed -s.\nThen let's prove that \n))\n2\n(\n)\n(\n)(\n1\n(\n)\n1,\n1\n("},{"paragraph_id":"p79","order":79,"text":"s\ns\nq\ns\nq"},{"paragraph_id":"p80","order":80,"text":".\nThe generalized Binet-Minc identity for \n)\n(\nˆ\nA\nB\nr\ne\np\n:\nLet A , B be an \nr\nn - and \nr\nm -matrix, then"},{"paragraph_id":"p81","order":81,"text":"I\nR\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nI\nI\nK\nn\nN\nK\nB\nA\nscal\nI\nB\nA\nscal\nA\nr\ne\np\n)\n(\n)\n(\n)\n(\n)\n)!\n1\n|\n(|\n(\n)\n(\n)1\n(\n)1\n(\n)\n(\nˆ\n)\n,\n(\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|)\n\"\n|\n1\n)(\n2\n|'\n(|\n|\n|\n3\n,\n2\n,1\n1\n;\n2\n,1\n)\n("},{"paragraph_id":"p82","order":82,"text":"where:\n},\n,...,\n1\n{\n},\n,...,\n1\n{\n},\n,...,\n1\n{\nr\nR\nm\nM\nn\nN"},{"paragraph_id":"p83","order":83,"text":"for two disjoint sets \n'\nH and \n\"\nH ,\n)\"\n,'\n(\n1\n;2\n,1\nH\nH\nPart"},{"paragraph_id":"p84","order":84,"text":"is the set of partitions of the set \n\"\n' H\nH \non sets having 1 or 2 elements in \n'\nH and at least 1 element in \n\"\nH (those sets are denoted \nhere as pairs \n\"\n\"\n,'\n'\n),\n\"\n,'\n(\nH\nI\nH\nI\nI\nI"},{"paragraph_id":"p85","order":85,"text":").\nProof:\nby analogy with the previous proof, we re-write the right part as"},{"paragraph_id":"p86","order":86,"text":"I\nR\nI\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nI\nI\nN\nK\nA\nscal\nI\nB\nA\nscal\n)\n(\n)\n(\n)\n(\n)1\n(\n)!\n1\n|\n(|\n)\n(\n)1\n(\n)\n,\n(\n1\n|\n|\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|)\n\"\n|\n1\n)(\n2\n|'\n(|\n)\n(\n3\n,\n2\n,1\n1\n;\n2\n,1"},{"paragraph_id":"p87","order":87,"text":"Let’s denote it by \n)\n,\n(ˆ\nB\nA"},{"paragraph_id":"p88","order":88,"text":"."},{"paragraph_id":"p89","order":89,"text":"Then, by the construction,\n)\n(\n}\n,...,\n{\n)\n(\n}\n,...,\n{\n}\n,...,\n1\n{\n,\n,\n'\n\"\n\"\n'\n|\n|\n|\n|\n\\\n\"\n|\n|\n\"\n1\n'\n|\n|\n'\n1\n\"\n1\n'\n1\n\"\n|\n|\n'\n|\n|\n'\n\"\n\"\n'\n)\n1,\n1\n(ˆ\n...\n)\n,\n(ˆ\nM\nPart\nI\nI\nN\nPart\nI\nI\nr\nJ\nI\nI\nI\nI\nJ\nj\nI\ni\nI\ni\nj\ni\nj\ni\nI\nI\nJ\nJ\nJ\nJ\nj\nj\nj\nj\nb\na\nB\nA"},{"paragraph_id":"p90","order":90,"text":"Then let's prove that \n)1\n(\n)\n1,\n1\n(ˆ"},{"paragraph_id":"p91","order":91,"text":"q\ns\nq"},{"paragraph_id":"p92","order":92,"text":"Lemma 1 (known, for any characteristic, Ref. [2], [6]):\nLet \n).\ndim(\n)\ndim(\ny\nx \n Then\n)\n,\n(\n))\n(\ndet(\n))\n(\ndet(\n)1\n(\n))\n,\n(\ndet(\n2\n)\n1\n)\n)(dim(\ndim(\ny\nx\npol\ny\nVan\nx\nVan\ny\nx\nC\nx\nx"},{"paragraph_id":"p93","order":93,"text":"Corollary 1.1 :\nLet \n).\ndim(\n)\ndim(\n2\ny\nx \n Then\n)\n,\n(\n))\n(\ndet(\n))\n(\n(\ndet\n)\n)\n,\n(\n)\n,\n(\ndet(\n2\n4\n2\ny\nx\npol\ny\nVan\nx\nVan\ny\nx\nC\ny\nx\nC"},{"paragraph_id":"p94","order":94,"text":"Proof:\n)\n(\n)\n(\n))\n,\n1\n(\ndet(\n)1\n(\nlim\n))\n)\n,\n1\n(\n)\n,\n(\ndet(\nlim\n)\n)\n,\n(\n)\n,\n(\ndet(\n)\ndim(\n)\ndim(\n)\ndim(\n0\n)\ndim(\n)\ndim(\n0\n2\ny\nx\nx\nC\ny\nx\nC\ny\nx\nC\ny\nx\nC\ny\nx\nC\nx\nx\nx\nx\nx"},{"paragraph_id":"p95","order":95,"text":"Lemma 2 (for any characteristic, Ref. [5]):\nLet A , B be \nm\nn \n-matrices, \nm\nn \n. Then \n(*) \nFor \n)\n2\n(mod\n0"},{"paragraph_id":"p96","order":96,"text":"n\n: \n)\n)}\n(\n{\ndet(\n)\n(\ndet2\nT\nm\nm\nA\nj\ni\nsign\nA\nA"},{"paragraph_id":"p97","order":97,"text":", \n)\n)}\n(\n({\n)\n)}\n(\n{\n(\n)\ndet(\nn\nn\nT\nm\nm\nj\ni\nsign\nPf\nA\nj\ni\nsign\nA\nPf\nA"},{"paragraph_id":"p98","order":98,"text":"where for a real"},{"paragraph_id":"p99","order":99,"text":"0\n,0\n0\n,1\n0\n,1\n)\n(\nu\nu\nu\nu\nsign\nu"},{"paragraph_id":"p100","order":100,"text":"For \n,\n,...,\n1\n,\nm\nq\nn"},{"paragraph_id":"p101","order":101,"text":"scalar :\n)\ndet(\n)\n0\n0\n0\n2\n1\n1\n0\n0\n0\ndet(\n2\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n2\n2\n)\n2\n(\n2\nA\nI\nI\nA\nq\nm\nq\nm\nq\nq\nm\nq\nm\nq\nq\nm\nq\nq\nq"},{"paragraph_id":"p102","order":102,"text":". \nIn the generic case this method allows to reduce A to a triangular matrix\nwhen computing \n)\ndet(A .\n(**) for \n)\n2\n(mod\n0"},{"paragraph_id":"p103","order":103,"text":"n"},{"paragraph_id":"p104","order":104,"text":")\n|\n|\ndet(\n)\n|\ndet(\n1\n)\n,\n(\n)\n,\n(\n)\n,\n(\n)\n,\n(\n2\n/\n|\n|,\n)\n,\n(\n)\n,\n("},{"paragraph_id":"p105","order":105,"text":"m\nk\nk\nN\nk\nN\nk\nN\nk\nN\nn\nJ\nJ\nJ\nj\nj\nN\nj\nN\nB\nA\nB\nA\nB\nA"},{"paragraph_id":"p106","order":106,"text":",\nwhere \n}\n,...,\n1\n{\nn\nN \n.\nLemma 3 (for any characteristic):\nLet \n)\n(x\nf\n be a polynomial in \n)\ndim(\n1,...,\nx\nx\nx\n of degree d . Then \n)))\n(\n(\n(\n)\n(\n0"},{"paragraph_id":"p107","order":107,"text":"x\nx\nf\ncoef\nf\nd\ni\ni\nwhere \ni\ncoef is the coefficient at \ni (if the expression is considered as a polynomial in ).\nThe neighbouring computation principle (for any characteristic):\nLet \n)\n(u\nf\n be a polynomial in \n)\ndim(\n1,...,\nu\nu\nu\n of degree d over the field F , \n)\n,...,\n(\n)\ndim(\n1\nh\nh\nh\nu\nu \n , u exists at \n]\n0\n[h\n;\n)\n,...,\n(\n)\ndim(\n1\nh\nh\nh\nv\nv \n, \n)\ndim(\n]\n0\n[\n0\n)\n(\nv\nh\nv"},{"paragraph_id":"p108","order":108,"text":";\n)\n,\n(\n]\n0\n[h\nv\nu"},{"paragraph_id":"p109","order":109,"text":"exists and is nonsingular. \nThen, given , over \n)\n( \nF\n: \n)))\n(\n(\n(\n)\n(\n0\n]\n[\n0"},{"paragraph_id":"p110","order":110,"text":"d\nk\nk\nk\nd\ni\nh\nu\nf\ncoef\nf\ni"},{"paragraph_id":"p111","order":111,"text":".\nwhere:\nd\nk\nh\nu\ncoef\nh\nu\nk\nh\nh\nv\nu\nv\nk\ni\ni\ni\nk\nk\n,...,\n1\n,\n0\n))\n(\n(\n)\n)\n(\n)(\n1\n(\n)\n,\n(\n)\ndim(\n1\n0\n]\n[\n]\n0\n[\n]\n[\n]\n0\n["},{"paragraph_id":"p112","order":112,"text":"(And thus \n)\n(\n)\n)\n(\n(\n)\n(\n)\n(\n1\n]\n0\n[\n]\n0\n[\n0\n]\n["},{"paragraph_id":"p113","order":113,"text":"d\nd\nk\nk\nk\nO\nh\nu\nh\nu\nh\nu"},{"paragraph_id":"p114","order":114,"text":")\nThis method will be called the \nng\nneighbouri\n computation of the function \n)\n( \nf\n by the\nzation\nparameteri\n)\n(h\nu\n in the region\n)\ndim(\n0\n)\n(\nv\nh\nv"},{"paragraph_id":"p115","order":115,"text":"from the \nbearing point \n]\n0\n[h\n."},{"paragraph_id":"p116","order":116,"text":"If \n))\n(\n(\nh\nu\nf\n is computable in polynomial time for any h in the region\n)\ndim(\n0\n)\n(\nv\nh\nv"},{"paragraph_id":"p117","order":117,"text":"and there exists a bearing point, then \n)\n( \nf\n is computable \nin polynomial time for any too. \nIn the further, to be clear and short, we’ll call a system of functions S \nalgebraically absolutely independent in a region R (given by a system \nof equations with a zero right part) iff the joint system of functions consisting \nof S and the left part of the system representing R is algebraically independent \nat some point of R. \nThe prolongation-derivative principle (for any characteristic):\na left (right) prolongation pattern of a vector-variety can be removed with an \ninduced coordinate-wise transformation of the remaining vector of extension-planes \nsuch that the E-sum preserves its value. Such a transformation will be called a left (right) \nprolongation-derivative, or the prolongation-derivative on the given prolongation. \nFormally it will be denoted (correspondingly for the left and right cases) by\n)\n)1,\n(\n)\n,\n(\n,\n(\n)\n)\n,\n,\n(\n,\n(\n)\n,\n(\n)\n,\n(\n)\n)\n,1(\n)\n,\n(\n,\n(\n)\n)\n,\n,\n(\n,\n(\n)\n,\n(\n)\n,\n(\n)\ndim(\n)\ndim("},{"paragraph_id":"p118","order":118,"text":"esum\nd\nd\nesum\nesum\nd\nd\nesum\nd\nd\nesum\nesum\nd\nd\ndef\nR\nR\ndef\nR\nR\ndef\nL\nL\ndef\nL\nL\nThis principle is based on the Binet-Minc identity."},{"paragraph_id":"p119","order":119,"text":"Lemma 4:\nLet A be an \nn\nn \n-matrix, B be a \nn\nn \n)\n2\n(\n-matrix. Then"},{"paragraph_id":"p120","order":120,"text":")\n(\ndet\n)\n|\n(\n)\n|\n(\n2 A\nB\nB\nper\nBA\nBA\nper"},{"paragraph_id":"p121","order":121,"text":"The Borchardt formula (for any characteristic, Ref. [2]) :\nLet \n)\ndim(\n)\ndim(\ny\nx \n. Then\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n))\n,\n(\n(\n2\ny\nx\nC\ny\nx\nC\ny\nx\nC\nper"},{"paragraph_id":"p122","order":122,"text":"Lemma 5:\n))\n(\ndet(\n)\ndet(\n))\n(\n(\n)\ndim(\nt\nVan\nt\nt\nW\nper\nt"},{"paragraph_id":"p123","order":123,"text":"where \n)\ndim(t"},{"paragraph_id":"p124","order":124,"text":"is the first \n)\ndim(t members of the sequence \ncomposed as the sequence of pairs"},{"paragraph_id":"p125","order":125,"text":",...,\n0\n),\n1\n3,\n3\n(\ns\ns\ns"},{"paragraph_id":"p126","order":126,"text":"Proof:"},{"paragraph_id":"p127","order":127,"text":")\ndim(\n1\n0\n))\n(\n(\n))\n,\n(\n(\nlim\n))\n(\n(\ny\nj\njt\nx\nW\nper\nt\nx\nC\nper\nt\nW\nper"},{"paragraph_id":"p128","order":128,"text":"The main part.\nTheorem I :\nLet \nn\nu\nt\n2\n)\ndim(\n2\n)\ndim("},{"paragraph_id":"p129","order":129,"text":".\nThen\nProof:\n))\n(\n(\ndet\n)\n,\n(\n))\n(\n(\n)\n,\n(\n)\n)\n(\n0\n0\n)\n(\n)\n(\n(\n)\n,\n(\n))\n,\n(\n)\n)}\n,\n(\n({\n(\n))\n,\n(\n(\n2\n2\n1\n2\nu\nR\nu\nt\npol\nt\nW\nper\nu\nt\npol\nu\nR\nu\nR\nt\nW\nper\nu\nt\npol\nu\nt\nC\nu\nt\npol\nDiag\nper\nu\nt\nC\nper\nT\nT\nn\ni\ni\nn\ni"},{"paragraph_id":"p130","order":130,"text":"where \n)\n(u\nR\n is the matrix of the coefficients of the polynomials\nin the formal scalar variable \n)\n,\n(\n),...,\n,\n(\n\\\n1\\\nn\nu\npol\nu\npol"},{"paragraph_id":"p131","order":131,"text":". The determinant \nof this matrix is a homogenous polynomial in \nn\nu\nu ,...,\n1\n of degree \n2\n)1\n(\n)1\n(\n...\n1\n0"},{"paragraph_id":"p132","order":132,"text":"n\nn\nn\n, and in the meantime it is divided by \nj\ni\nu\nu \nfor any \nn\nj\ni"},{"paragraph_id":"p133","order":133,"text":"1\n, or by \n))\n(\ndet(\nu\nVan\n; moreover, its coefficients \nare 1 and -1, therefore \n))\n(\n(\ndet\n))\n(\n(\ndet\n2\n2\nu\nVan\nu\nR"},{"paragraph_id":"p134","order":134,"text":", what completes \nthe proof.\nCorollary I.1 :\nLet \n)\ndim(\n)\ndim(\n2\n)\ndim(\nv\nu\nt"},{"paragraph_id":"p135","order":135,"text":".Then\n)\n,\n(\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n))\n,\n(\n(\n2"},{"paragraph_id":"p136","order":136,"text":"v\nu\nt\npol\nv\nu\npol\nu\nVan\nv\nt\nW\nper\nv\nu\nu\nt\nC\nper\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n))\n,\n(\n(\n2\nu\nt\npol\nu\nVan\nt\nW\nper\nu\nu\nt\nC\nper"},{"paragraph_id":"p137","order":137,"text":"Proof:\n))\n,\n1\n(\n(\nlim\n))\n,\n(\n(\n)\ndim(\n)\ndim(\n0"},{"paragraph_id":"p138","order":138,"text":"v\nv\nu\nu\nv\nt\nC\nper\nv\nu\nu\nt\nC\nper\nv\nv"},{"paragraph_id":"p139","order":139,"text":"and, basing on Theorem I, we calculate the limit to get the lemma’s equality.\nCorollary I.2 :\nLet \n)\ndim(\n)\ndim(\n2\n)\ndim(\nv\nu\nt"},{"paragraph_id":"p140","order":140,"text":"+1. Then\nProof:\n)\n/\n))\n/\n1\n,\n(\n(\n(\nlim\n)\n1\n)\n,\n(\n(\n0\n)\ndim("},{"paragraph_id":"p141","order":141,"text":"v\nu\nu\nt\nC\nper\nv\nu\nu\nt\nC\nper\nt"},{"paragraph_id":"p142","order":142,"text":"and then, basing on Corollary I.1, we again just calculate the limit.\nTheorem II :\n)\n)\n(\n)\n(\n0\n)\n('\n)\n(\n)\n('\n)\n(\ndet(\n)\n,\n,\n(\n)\ndim(\n1\n]\n[\n]\n[\n]\n[\n]\n[\n]\n[\n]\n[\n,"},{"paragraph_id":"p143","order":143,"text":"u\ni\ni\nq\np\ni\nq\np\nq\np\ni\nq\np\ni\ni\nq\np\ni\ni\nq\np\ni\ni\nq\np\ni\nq\np\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\ndis"},{"paragraph_id":"p144","order":144,"text":"Proof:\nbasing on Lemma 2, we conclude that the right part of this equality is"},{"paragraph_id":"p145","order":145,"text":"p\nI\nI\nI\ni\ni\nq\np\nq\np\ni\nq\np\ni\ni\nq\np\ni\nu\nvan\nu\nvan\nu\nvan\n|\n|,\n]\n[\n]\n[\n]\n[\n)\n)\n(\n0\n)\n('\n)\n(\ndet("},{"paragraph_id":"p146","order":146,"text":"Theorem III :\nLet\n)\n,\n(\n)\n,\n('\n,0\n)\n,\n(\"\n,1\n3\n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n|\n|"},{"paragraph_id":"p147","order":147,"text":"y\nx\npol\nt\npol\ny\nx\npol\nn\nt\nn\ny\nn\nx\nI"},{"paragraph_id":"p148","order":148,"text":"Then\n)\n,\n(\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n)\n1\n)\n,\n(\n(\n2\n)\ndim("},{"paragraph_id":"p149","order":149,"text":"v\nu\nt\npol\nv\nu\npol\nu\nVan\nv\nt\nW\nper\nv\nu\nu\nt\nC\nper\nt"},{"paragraph_id":"p150","order":150,"text":"I\ni\ni\nI\nI\ny\ny\npol\ny\nVan\nt\nx\npol\nx\nt\nW\nper\ny\nx\nC\nper\n)\n,\n('\n1\n))\n(\n(\ndet\n)\n,\n(\n))\n(\n(\n))\n,\n(\n(\n2\nProof:\nLet us apply the Binet-Minc identity to \n)\n)\n1\n|\n)\n,\n(\n((\n)\ndim(\nT\nt\nI\nI\ny\ny\nx\nt\nC\nper"},{"paragraph_id":"p151","order":151,"text":"Theorem IV :\nLet\n)\n,\n(\n)\n,\n('\n,0\n)\n,\n(\"\n)\n2\n(mod\n0\n)\ndim(\n,1\n3\n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim("},{"paragraph_id":"p152","order":152,"text":"y\nx\npol\nt\npol\ny\nx\npol\nm\nz\nn\nt\nn\ny\nn\nx"},{"paragraph_id":"p153","order":153,"text":"Then"},{"paragraph_id":"p154","order":154,"text":"))\n(\n)\n,\n(\n),\n(\n)\n,\n(\n("},{"paragraph_id":"p155","order":155,"text":"Diag\ny\nz\nC\nDiag\ny\nx\nC\ncoper\n)\n}\n)\n,\n(\n{,\n}\n)\n,\n('\n{,\n(\n)\n,\n(\n))\n(\n(\n))\n(\n(\n2\n2\n2\n2\n/\n,\nn\ni\ni\nn\ni\ni\nm\nn\nz\ny\npol\ny\ny\npol\ny\ndis\nt\nx\npol\nz\nW\nper\nx\nt\nW\nper"},{"paragraph_id":"p156","order":156,"text":"Proof:\nIt follows from the definition of the copermanent and Theorems I, III. \n----------------------------------------------------------------------------------------\nLet’s note that for \n0\n)\n,\n(\""},{"paragraph_id":"p157","order":157,"text":"y\nx\npol \n to be true it’s sufficient that \n0\n)\n,\n(\""},{"paragraph_id":"p158","order":158,"text":"y\nx\npol \n for any n pair-wise distinct values of (due to the characteristic), \nfor instance for all the coordinates of x , i.e. it’s equivalent to \nn\nn\ni\ny\nx\nx\npol\n0\n)}\n,\n(\"\n{"},{"paragraph_id":"p159","order":159,"text":",\nor \nn\nn\nn\nn\ni\ni\ny\nx\nC\nx\nC\ny\nx\nx\npol\ny\nx\nx\npol\n0\n1\n)\n,\n(\n1\n)\n(\n~\n)}\n,\n('\n/)\n,\n(\"\n{\n2"},{"paragraph_id":"p160","order":160,"text":".\nTheorem V :\nLet\n),\n2\n(mod\n0\n)\ndim(\n,\n)\ndim("},{"paragraph_id":"p161","order":161,"text":"m\nz\nn\nx\n3,2\n,\n1\n)\n,\n("},{"paragraph_id":"p162","order":162,"text":"h\ny\nx\nC\nn\nh\nh"},{"paragraph_id":"p163","order":163,"text":". \nThen"},{"paragraph_id":"p164","order":164,"text":"n\ns\ns\nm\nz\nz\ny\nx\nC\nDiag\ny\nz\nC\nDiag\ny\nx\nC\ncoper\ny\nz\nC\ny\nz\nC\ny\nx\nC\nx\nz\ngen\n1\n)\n2\n/\n1\n(\n2\n/\n)\ndim(\n3\n3\n)\ndim(\n2\n2\n)\n1\n(\n*\n)\n,\n(\n))\n(\n)\n,\n(\n),\n(\n)\n,\n(\n(\n)1\n(\n)\n1\n)\n,\n(\n,\n1\n)\n,\n(\n,\n)\n)\n,\n(\n(,\n,\n("},{"paragraph_id":"p165","order":165,"text":"Proof:\nIt follows from the Binet-Minc identity for \n)\n,\n(\nB\nA\ncoper\nTheorem VI :\n Let\nn\nm\nz\nn\ny\nn\nx"},{"paragraph_id":"p166","order":166,"text":")\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n. \nThen there exist scalar constants \n3\n2,"},{"paragraph_id":"p167","order":167,"text":"such that the system of functions in"},{"paragraph_id":"p168","order":168,"text":",\n, y\nx\n($)"},{"paragraph_id":"p169","order":169,"text":"3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n3,2\n,\n)\n,\n(\n:\n,..,\n1\n)\n(\nq\ns\nq\ny\nx\nC\nx\nz\nC\ns\ny\nz\nC\nm\nk\ns\nk\nq\ns\nk"},{"paragraph_id":"p170","order":170,"text":"is algebraically absolutely independent in the region \n)\n("},{"paragraph_id":"p171","order":171,"text":"3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nn\ns\ns\nn\nn\nn"},{"paragraph_id":"p172","order":172,"text":"Proof:\nlet us consider the Jacobian matrix which is"},{"paragraph_id":"p173","order":173,"text":")\n,\n)}\n,\n(\"\n{\n)\n)\n,\n(\n)(\n,\n(\n)\n)\n,\n(\n)(\n,\n(\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n)\n,\n(\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n)\n2\n,1(\n)\n1\n(\n3\n2"},{"paragraph_id":"p174","order":174,"text":"y\nx\ny\nx\nx\npol\ny\nx\nC\nx\nz\nC\ny\nx\nC\nx\nz\nC\ny\nx\nC\nx\nz\nC\ny\nz\nx\nC\ny\nz\nx\nC\nn\ni"},{"paragraph_id":"p175","order":175,"text":")\n2\n(\n2\n2\n2\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n3\n(\n3\n)\n3\n(\n2\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n)\n2\n(\n)\n(\n3\n2\n)\n2\n(\n3\n2\n0\n)\n,\n(\n)\n1\n)\n,\n(\n1\n)\n(\n~\n(\n)\n(\n~\n0\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)\n(\n)\n,\n(\n0\n)\n)\n,\n(\n(\n)\n(\n)\n,\n(\nn\nn\nn\nn\nn\nm\nn\nm\nn\nn\nm\nn\nm\ny\nx\nC\ny\nx\nC\nx\nC\nDiag\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\ny\nx\nC\nDiag\nDiag\ny\nz\nx\nC"},{"paragraph_id":"p176","order":176,"text":"where -s denote some matrices of appropriate dimensions. For the purpose of \nsimplicity, let’s permute in a certain way the Jacobian’s rows and multiply some \nof them by -1, thus reducing it to the form"},{"paragraph_id":"p177","order":177,"text":"22\n21\n)\n2\n(\n)\n2\n3\n(\n11\n0\nA\nA\nA\nn\nn\nm"},{"paragraph_id":"p178","order":178,"text":"where"},{"paragraph_id":"p179","order":179,"text":")\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n22\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\nA"},{"paragraph_id":"p180","order":180,"text":"If both the matrices are nonsingular then the Jacobian matrix is nonsingular too. \nIf F is the basic field, let’s consider them over \n)\n( \nF\n for \n)).\ndim(\n(mod\n0\n)\ndim(\n,\nover"},{"paragraph_id":"p181","order":181,"text":"vectors\nare\n )\n(\n,\n,\n,"},{"paragraph_id":"p182","order":182,"text":"where\n,\n,\n1\n,\n,\n1\n,\n]\n[\n]\n[\n]\n1\n[\n0\n]\n[\n)\ndim(\n/)\ndim(\n2\n]\n0\n[\n0\n]\n[\n)\ndim(\n/)\ndim(\n]\n0\n[\n]\n1[\n]\n0\n[\nx\nx\nF\nk\ny\nx\nh\nh\ny\ny\ny\nh\nx\nx\nx\nx\nk\nk\nk\nk\nk\nh\nx\nk\nk\nk\nh\nx"},{"paragraph_id":"p183","order":183,"text":"First we’ll show how an -point (satisfying the region’s conditions \n)\n( )\nof such a type is built and then we’ll prove that in the generic case it gives \nnonsingular \n11\nA ,\n22\nA .\nIn order to build such an -point, it’s sufficient that the Jacobian matrix of"},{"paragraph_id":"p184","order":184,"text":")\n(\n)\n,\n(\n0\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n1\n)\n,\n(\n1\n)\n(\n~\n(\n)\n(\n~\n)\n(\n)\n,\n(\n)\n)\n,\n(\n(\n3\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n3\n(\n)\n3,2\n(\n2\n2\n2\n2\n2\n3\n2\n3\n2\n11"},{"paragraph_id":"p185","order":185,"text":"Diag\ny\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nx\nC\nDiag\nx\nC\nDiag\ny\nx\nC\ny\nx\nC\nDiag\nA\nn\nm\nn\nm\nn\nn"},{"paragraph_id":"p186","order":186,"text":"the system of functions representing the left part of \n)\n( exists and is nonsingular\nat a certain approximation of the power series (in ) for"},{"paragraph_id":"p187","order":187,"text":",y\nwhich satisfies\nthe region’s conditions up to some degree of (then all the other members of \nthe power series are computable via corresponding linear equations involving \nthe Jacobian matrix calculated for the approximation). Let’s consider only the first \ntwo members of the power series for y (i.e. \n]\n0\n[y\n+"},{"paragraph_id":"p188","order":188,"text":"]\n1\n[y\n) and the first member of the \none for (i.e. \n]\n0\n[ \n).\nFor \n11\nA , let’s multiply its first two block-rows by \n2\n and then consider its\n0\nlim"},{"paragraph_id":"p189","order":189,"text":". In this limit let’s substitute the matrix received via summing up all \nthe columns corresponding to the same \nrh in \n)\ndim(\n,...,\n1\n,\n]\n0\n[\nh\nr\ny"},{"paragraph_id":"p190","order":190,"text":", for its \nfirst block-column (such a transformation can’t enlarge the rank) .\nAnd now \n22\nA :"},{"paragraph_id":"p191","order":191,"text":")\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n(\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n22\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nC\nDiag\ny\nx\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\nA"},{"paragraph_id":"p192","order":192,"text":"P\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nC\ny\nx\nC\nDiag\ny\nx\nC\nP\nI\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\ny\nz\nC\ny\nx\nC\nDiag\ny\nx\nC\nP\nDiag\ny\nx\nC\nn\nm\nn\nm\nn\nm\ni\nj\nj\nn\nn\nm\nn\nn\nn\nn\nn\nm\ni\nj\nj\nn"},{"paragraph_id":"p193","order":193,"text":")\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)}\n)\n(\n)\n,\n(\n)\n,\n(\n)\n(\n)\n,\n(\ndet(\n)\ndet(\n)1\n(\n{\n0\n0\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)}\n)\n(\n)\n,\n(\n)\n,\n(\n)\n(\n)\n,\n(\ndet(\n)\ndet(\n)1\n(\n{\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n\\\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n\\"},{"paragraph_id":"p194","order":194,"text":"where"},{"paragraph_id":"p195","order":195,"text":")\n,\n(\n)\n(\n)\n,\n(\ny\nx\nC\nDiag\ny\nx\nC\nP"},{"paragraph_id":"p196","order":196,"text":".\nSince we use the expression \n)\ndet(\n1\nP , first let’s prove the non-singularity of P\nin the generic case. It’s clearly seen for \n)\ndim(\n)\ndim(\nx\nh \n. \nCorollary VI.1 :\n)\n,\n,\n,\n,\n(\n]\n2\n[\n]\n1["},{"paragraph_id":"p197","order":197,"text":"u\nz\ngen \n (which, according to the Binet-Minc identity, \nis a function in the essential variables: the Cauchy - Binet - Minc weights\n)\n3\n,...,\n,2,1\n(\n)\n,\n(\nq\ns\nq\nu\nz\nC\ns\ni\nq"},{"paragraph_id":"p198","order":198,"text":"and the star - functional weights\n]\n2\n[\n]\n1\n[ ,\ni\ni"},{"paragraph_id":"p199","order":199,"text":")"},{"paragraph_id":"p200","order":200,"text":"is computable in polynomial time for any values of the essential (and, hence, \noriginal) variables via a neighbouring computation based on Theorems V, VI \nas the following:\nthe parameterization (of the essential variables) in"},{"paragraph_id":"p201","order":201,"text":",\n, y\nx\n:"},{"paragraph_id":"p202","order":202,"text":"3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n2,1\n,\n1\n)\n,\n(\n)\n2\n(mod\n0\n)\ndim(\n)\n(\n1\n1\n]\n[\nq\ns\nq\ny\nx\nC\nx\nz\nC\nu\nz\nC\ns\ny\nz\nC\nm\nz\ns\ns\nq\nm\ns\ns\ns"},{"paragraph_id":"p203","order":203,"text":"the region :"},{"paragraph_id":"p204","order":204,"text":"3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\n)\ndim(\n2\n)\ndim(\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nx\ny\nn\nn\ns\ns\nn\nn\nn"},{"paragraph_id":"p205","order":205,"text":"Hence the final formula for \n)\n,\n,\n,\n,\n(\n]\n2\n[\n]\n1["},{"paragraph_id":"p206","order":206,"text":"u\nz\ngen \n is:\n)\n}\n)\n,\n(\n{,\n}\n)\n,\n('\n{,\n(\n)\n,\n(\n))\n(\n(\n))\n(\n(\n)\n,\n,\n,\n,\n(\n2\n2\n2\n/\n,\n]\n2\n[\n]\n1[\nn\ni\ni\nn\ni\ni\nm\nn\nz\ny\npol\ny\ny\npol\ny\ndis\nt\nx\npol\nz\nW\nper\nx\nt\nW\nper\nu\nz\ngen"},{"paragraph_id":"p207","order":207,"text":"where: \n)\n2\n(mod\n0\n)\ndim(\n,\n2\n)\ndim(\n2\n)\ndim(\n,1\n3\n)\ndim("},{"paragraph_id":"p208","order":208,"text":"m\nz\nn\nx\ny\nn\nt\n ,"},{"paragraph_id":"p209","order":209,"text":")\n,\n(\n)\n,\n('"},{"paragraph_id":"p210","order":210,"text":"y\nx\npol\nt\npol"},{"paragraph_id":"p211","order":211,"text":",\n the region :"},{"paragraph_id":"p212","order":212,"text":"3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nn\ns\ns\nn\nn\nn"},{"paragraph_id":"p213","order":213,"text":"the parameterization :"},{"paragraph_id":"p214","order":214,"text":"3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n2,1\n,\n1\n)\n,\n(\n)\n(\n1\n)1\n(\n]\n[\nq\ns\nq\ny\nx\nC\nx\nz\nC\nu\nz\nC\ns\ny\nz\nC\ns\nq\ns\nq\nm\ns\ns\ns"},{"paragraph_id":"p215","order":215,"text":"(\n3\n2,"},{"paragraph_id":"p216","order":216,"text":"are scalar constants)"},{"paragraph_id":"p217","order":217,"text":"Theorem VII :\n))\n0\n0\n1\n1\n0\n0\n,\n0\n1\n0\n0\n1\n1\n,\n1\n1\n1\n)\n/\n1(\n,\n,\n1\n1\n1\n1\n1\n(\n(\nlim\n)\n,\n,\n,\n1\n1\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n*\n)\ndim(\n)\ndim(\n)\ndim(\n2\n0\n)\ndim(\n)\ndim(\n~"},{"paragraph_id":"p218","order":218,"text":"w\nw\nv\nv\nu\nu\nw\nw\nv\nv\nu\nu\nv\nv\nu\nw\nv\nv\nu\nu\nw\nv\nu\nv\nu\nt\nv\nv\nu\nh\nh\nv\nv\nu\nu\ngen\nt\nh\nv\nu\ngen"},{"paragraph_id":"p219","order":219,"text":"Theorem VIII :\n))\n,\n,\n,\n1\n1\n,\n1\n(\n(\nlim\n)\n,\n,\n,\n,\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n0"},{"paragraph_id":"p220","order":220,"text":"t\nh\nw\nw\nz\ngen\nt\nh\nw\nz\ngen\nu\nu\nu"},{"paragraph_id":"p221","order":221,"text":"Theorem IX :\n)\n(\n))\n,\n,\n,\n,\n,\n(\n(\nlim\n)\n(\n)\n,\n,\n,\n0\n,\n(\n)\ndim(\n)\ndim(\n2\n0\n)\ndim(\n1\n)\ndim(\n~"},{"paragraph_id":"p222","order":222,"text":"y\nt\nx\nh\nw\nz\ngen\ncoef\nt\nh\nw\nz\ngen\nh\nw\nx\nj\nj\nu"},{"paragraph_id":"p223","order":223,"text":"where is a formal scalar variable, and for \n3,2,1\n,2,1"},{"paragraph_id":"p224","order":224,"text":"q\ns\n:\n(i)"},{"paragraph_id":"p225","order":225,"text":")\ndim(\n)\ndim(\n)\ndim(\n3\n1\n0\n)1\n(\n)\n2\n(\n)\n(\n)1\n(\n)\n,\n(\n)\n,\n(\n)\n(\nz\nt\nw\ns\ns\nq\ns\nq\ns\nq\ns\nx\nt\nz\nC\nx\nw\nC"},{"paragraph_id":"p226","order":226,"text":"(ii)"},{"paragraph_id":"p227","order":227,"text":")\n1\n(\n)\ndim(\n)\ndim(\n)\ndim(\n3\n)1\n(\n0\n)1\n(\n)\n2\n(\n)\n(\n)\n,\n(\n)\n,\n(\n)\n,\n(\n)1\n(\n)\n("},{"paragraph_id":"p228","order":228,"text":"q\nq\ns\nq\ns\ny\nx\nC\ny\nh\nz\nC\ny\nw\nC\nz\nh\nw\ns\nq\ns\ns"},{"paragraph_id":"p229","order":229,"text":"Proof:\nAccording to the definition,"},{"paragraph_id":"p230","order":230,"text":"))\n,\n,\n,\n,\n,\n(\n(\n)\ndim("},{"paragraph_id":"p231","order":231,"text":"y\nt\nx\nh\nw\nz\ngen\ncoef\nh"},{"paragraph_id":"p232","order":232,"text":"I\nJ\nJ\nw\nI\nJ\nI\nJ\nI\ny\nK\nx\nh\nJ\nI\nK\nh\nK\nK\nw\nz\nesum\nw\nz\ny\nt\nx\nh\nesum\n)\n)\n0,0\n(\n,\n(\n)\n)\n0,0\n(\n)\n0,\n(\n)\n,\n(\n)\n0,\n(\n)\n,0\n(\n)\n,0\n(\n,\n(\n|\n|\n)\ndim(\n\\\n\\\n\\\n|\n|\n|\n|\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n|\n|,"},{"paragraph_id":"p233","order":233,"text":"and in such a case \n)\n)\n0,\n(\n)\n,\n(,\n(\n)\ndim("},{"paragraph_id":"p234","order":234,"text":"y\ny\n is a right prolongation. Then, according to the\nprolongation-derivative principle and the equality (ii), the first multiplier of the expression \nunder the summation signs is"},{"paragraph_id":"p235","order":235,"text":")\n)\n0,\n(\n)\n0,0\n,\n(\n)\n0,0\n,0\n(\n)1,\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n|\n|\n3\n2\n1\n1\n|\n|\n)\ndim(\n)\n1\n(\n)\ndim(\n)\n(\nJ\nI\nK\nx\nh\nJ\nI\nK\nw\nz\nt\nx\nh\nesum"},{"paragraph_id":"p236","order":236,"text":")\n)\n0,\n(\n)\n0,0\n,\n(\n)\n0,0\n,0\n(\n)1,\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n1\n|\n|\n3\n2\n1\n1\n|\n|\n)\ndim(\n)\ndim(\n)\ndim(\n1\n)\n("},{"paragraph_id":"p237","order":237,"text":"J\nI\nK\nx\nh\nJ\nI\nK\nx\nj\nj\nw\nz\nt\nx\nh\nesum"},{"paragraph_id":"p238","order":238,"text":"Then \n)\n)\n,0\n(\n)\n,\n(,\n(\n)\ndim("},{"paragraph_id":"p239","order":239,"text":"x\nx\n is a left prolongation, and we can apply the \nprolongation-derivative principle again, this time together with the equality (i), \nhence receiving"},{"paragraph_id":"p240","order":240,"text":")\ndim(\n1\n1\nx\nj\nj"},{"paragraph_id":"p241","order":241,"text":")\n)\n0,0\n,\n(\n)\n0,0\n(\n)\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n|\n|\n2\n2\n1\n|\n|\n)\ndim(\n)\n("},{"paragraph_id":"p242","order":242,"text":"J\nI\nK\nh\nJ\nI\nK\no\nw\nz\nt\nh\nesum"},{"paragraph_id":"p243","order":243,"text":"Because the second multiplier doesn’t depend on , we can apply the operator \n...)\n(\nlim\n)\ndim(\n2\n0\nw"},{"paragraph_id":"p244","order":244,"text":"to the first one only, hence receiving\n))\ndim(\n|\n(|\n)\n)\n0,0\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n))\ndim(\n|\n(|\n)\n)\n0,0\n,\n(\n)\n0,0\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n)\ndim(\n1\n)\ndim(\n|\n|\n)\ndim(\n)\ndim(\n1\n)\ndim(\n|\n|\n)\ndim(\n)\n(\nw\nJ\nw\nz\nt\nh\nesum\nw\nJ\nw\nz\nt\nh\nesum\nx\nj\nj\nw\nI\nK\nh\nI\nK\nx\nj\nj\nw\nI\nK\nh\nI\nK"},{"paragraph_id":"p245","order":245,"text":"where, taking into account that the third sum"},{"paragraph_id":"p246","order":246,"text":")\ndim(\n|\n|,\nh\nK\nK\nruns over sets of cardinality \n)\ndim(h (thus “balancing” the patterns \n)\ndim(\n)\n,0\n(\nh"},{"paragraph_id":"p247","order":247,"text":"and \nK"},{"paragraph_id":"p248","order":248,"text":")\n0,\n( \n), the last passage \nis due to the requirement (following from the E-sum’s definition) that the total"},{"paragraph_id":"p249","order":249,"text":"balance is to be 0. So, after subjecting the product of the two multipliers to the \noverall summation"},{"paragraph_id":"p250","order":250,"text":"I\nJ\nh\nK\nK\n)\ndim(\n|\n|,\n, we get the theorem’s claim. \nTheorem X (for any characteristic) :\n1)"},{"paragraph_id":"p251","order":251,"text":"P\nj\ni\nj\ni\nz\nPart\nP\nz\nz\nz\nC\nper\n}\n,\n{\n2\n)})\ndim(\n,...,\n1\n({\n)\n(\n1\n))\n(\n~\n(\n2\n ,\nwhere for a set N\n)\n(\n2 N\nPart\nis the set of its perfect matchings. \n2) Let \n)\ndim(\n)\ndim(\ny\nx \n. Then\n)\n(\n)\n(\n)\ndim(\n1\n)\n,\n(\n)\n,\n(\n)\n(\n~\n))\n,\n(\n(\n))\n,\n,\n(\n~\n(\n)\n(\nz\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nz\ny\nx\nC\nper"},{"paragraph_id":"p252","order":252,"text":"Proof:\n1) first let’s prove that, if for a \nd\nd \n-matrix A we define"},{"paragraph_id":"p253","order":253,"text":"d\nH\nd\ni\ni\ni\ndef\na\nA\nham"},{"paragraph_id":"p254","order":254,"text":"1\n)\n(\n,\n)\n(\nwhere \nd\nH is the set of d -permutations with one cycle, \n0\n))\n(\n~\n("},{"paragraph_id":"p255","order":255,"text":"u\nC\nham\nwhen \n2\n)\ndim("},{"paragraph_id":"p256","order":256,"text":"u\n.\nSuppose \n2\n)\ndim("},{"paragraph_id":"p257","order":257,"text":"u\n. Let’s partition the set \nd\nH on disjoint subsets consisting of d -1 \nd -cycles which differ from each other only in the position of the element d . Each of those \nsubsets can be received via taking a ( d -1)-cycle \n)\n,...,\n(\n1\n1"},{"paragraph_id":"p258","order":258,"text":"dh\nh\nh\n with elements from the set \n}\n1\n,...,\n1\n{"},{"paragraph_id":"p259","order":259,"text":"d\n and alternate placing d between neighbors in h (altogether there are d -1 options).\nIn such a case the sum of the corresponding d -1 summands in the expression \n))\n(\n~\n(\nu\nC\nham\n will be"},{"paragraph_id":"p260","order":260,"text":"1\n1\n1\n1\n1\n1\n1\n1\n0\n)\n1\n1\n(\n)\n1\n(\n1\n)\n)(\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\nd\ni\nd\nk\nd\nh\nd\nh\nd\ni\nh\nh\nh\nh\nd\nk\nh\nd\nd\nh\nh\nh\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nk\nnext\nk\ni\nnext\ni\ni\nnext\ni\nk\nnext\nk\nk\nnext\nk\nwhere"},{"paragraph_id":"p261","order":261,"text":"1\n,1\n1\n,1\n)\n(\nd\nt\nd\nt\nt\nt\nnext\n . \nTaking into account that for an \nn\nn \n-matrix A"},{"paragraph_id":"p262","order":262,"text":"})\n,...,\n1\n({\n)\n,\n(\n)\n(\n)\n(\nn\nPart\nP\nP\nI\nI\nI\nA\nham\nA\nper"},{"paragraph_id":"p263","order":263,"text":"and \n)\n(\n~\n))\n(\n~\n(\n)\n,\n(\nI\nI\nI\nz\nC\nz\nC"},{"paragraph_id":"p264","order":264,"text":", we get the first claim of the theorem;\n2) we apply the induction on \n)\ndim(z . \nFirst let’s prove the induction’s basis for \n1\n)\ndim("},{"paragraph_id":"p265","order":265,"text":"z\n. Let z be scalar. Then"},{"paragraph_id":"p266","order":266,"text":")\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n1\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n(\nlim\n)))\n,\n(\n(\n1\n))\n,\n(\n(\n(\nlim\n))\n,\n,\n(\n~\n(\n2\n2\n0\n0\ny\nx\nC\ny\nx\nC\nz\ny\nz\nx\nC\nz\ny\nz\nx\nC\ny\nx\nC\nper\nz\ny\nz\nx\nC\nper\nz\ny\nx\nC\nper"},{"paragraph_id":"p267","order":267,"text":")\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n1\n))\n(\n1\n))\n,\n(\n)\n,\n(\n(\n1\n))(\n,\n(\ndet(\n1\n)1(\n))\n,\n(\ndet(\n1\n(\nlim\n2\n2\n)\ndim(\n2\n2\n0\ny\nx\nC\ny\nx\nC\nO\ny\nz\nC\nx\nz\nC\ny\nx\nC\nO\ny\nx\nC\nx"},{"paragraph_id":"p268","order":268,"text":"(the second passage is due to the Borchardt formula).\nAnd now the induction’s step:\nlet \n1\n,\n)\ndim("},{"paragraph_id":"p269","order":269,"text":"m\nz\nm\nz\nis a scalar. Then"},{"paragraph_id":"p270","order":270,"text":")\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\n1\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\nlim\n)\n))\n,\n,\n(\n~\n(\n1\n))\n,\n,\n(\n~\n(\n(\nlim\n))\n,\n,\n(\n~\n(\n)\ndim(\n)\ndim(\n1\n1\n1\n1\n0\n1\n1\n0\n1\nz\nz\nm\nm\nm\nm\nm\nm\nm\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nz\ny\nz\nC\nz\nx\nz\nC\nDiag\nz\nC\nper\nz\ny\nz\nx\nC\nper\nz\ny\nx\nC\nper\nz\nz\ny\nz\nx\nC\nper\nz\nz\ny\nx\nC\nper"},{"paragraph_id":"p271","order":271,"text":")\n(\n(\n)\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\n1\n))\n(\n))\n0\n1\n))\n,\n(\n)\n,\n(\n(\n(\n)\n(\n~\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n(\n1\n))\n,\n(\n)\n,\n(\n(\n1\n))(\n,\n(\n(\n1\nlim\n)\ndim(\n2\n)\ndim(\n1\n)\ndim(\n2\n)\ndim(\n1\n1\n0\nz\nz\nm\nz\nx\nm\nm\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nO\ny\nz\nC\nx\nz\nC\nDiag\nz\nz\nC\nper\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\nO\ny\nz\nC\nx\nz\nC\ny\nx\nC\nper"},{"paragraph_id":"p272","order":272,"text":"The last passage is due to the part (1) of this theorem.\nTheorem XI :\n))\n,\n,\n,\n1\n1\n0\n,\n1\n1\n1\n1\n1\n1\n(\n(\nlim\n)\n,\n,\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n)\ndim(\n2\n0\n^"},{"paragraph_id":"p273","order":273,"text":"t\nh\nu\nu\nu\ngen\nt\nh\nu\ngen\nu\nu\nu\nu\nu\nu"},{"paragraph_id":"p274","order":274,"text":"Proof:"},{"paragraph_id":"p275","order":275,"text":")\nX(2)\nTheorem\nuse\n(we\n))\n,\n,\n(\n~\n(\n))\n(\n~\n(\n)\n1\n(\n)\n,\n,\n,\n1\n1\n,\n(\n))\n,\n,\n,\n1\n1\n0\n,\n1\n1\n1\n1\n1\n1\n(\n(\nlim\n)\ndim(\n|\n|,\n\\\n|\n|\n)\ndim(\n~\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n)\ndim(\n2\n0"},{"paragraph_id":"p276","order":276,"text":"I\nJ\nj\nJ\nh\nJ\nJ\nI\nJ\nI\nI\nu\nu\nu\nu\nu\nu\nu\nu\nt\nh\nC\nper\nu\nC\nper\nt\nh\nu\ngen\nt\nh\nu\nu\nu\ngen"},{"paragraph_id":"p277","order":277,"text":"J\nj\nJ\nJ\nh\nJ\nJ\nu\nJ\nI\nJ\nj\nJ\nh\nJ\nJ\nJ\nu\nJ\nI\nI\nI\nI\nI\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nDiag\nper\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nDiag\nu\nC\nper\nu\nC\nper"},{"paragraph_id":"p278","order":278,"text":"))\n,\n(\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n(\n)\nX(1)\nTheorem\nuse\nwe\n(\n))\n,\n(\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n(\n~\n(\n)\n1\n(\n)\ndim(\n|\n|,\n)\ndim(\n)\ndim(\n|\n|,\n)\ndim(\n\\\n\\\n\\\n|\n|"},{"paragraph_id":"p279","order":279,"text":")\n,\n,\n,\n(\n))\n,\n(\n(\n))\n1\n)\n,\n(\n1\n)\n,\n(\n(\n(\n^\n)\ndim(\n1\n)\ndim(\n)\ndim(\n)\ndim(\n|\n|,"},{"paragraph_id":"p280","order":280,"text":"t\nh\nu\ngen\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nu\ni\nJ\nj\nJ\nJ\nu\nJ\ni\nu\ni\nh\nJ\nJ"},{"paragraph_id":"p281","order":281,"text":"Theorem XII :\nLet \n.\n0\n)\n(\n)\n,\n(\n),\ndim(\n)\ndim(\n)\ndim(\n2\n)\n1,\n3\n1\n(\nh\nDiag\nt\nh\nC\nh\nu"},{"paragraph_id":"p282","order":282,"text":"Then\n)\n1\n,\n,\n,\n1\n(\n)\n1\n1(\n))\n,\n(\n)\n(\n)\n,\n(\n(\n3\n^\n)\ndim(\n3"},{"paragraph_id":"p283","order":283,"text":"t\nt\nh\nu\ngen\nh\nt\nC\nDiag\nt\nu\nC\nper\nm\nm\nh"},{"paragraph_id":"p284","order":284,"text":"Proof:\nIt follows from the generalized Binet-Minc identity for \n)\n(\nˆ\nA\nr\ne\np\nB\n, due to the fact \nthat \n0\n)\n1\n(\n)\n,\n(\n,3\n|\n|,\n)\n(\n)})\ndim(\n2\n,...,\n1\n{,\n()\n("},{"paragraph_id":"p285","order":285,"text":"t\nI\nDiag\nt\nt\nh\nC\nscal\nI\nI"},{"paragraph_id":"p286","order":286,"text":"(by the construction\nand the condition \n)\ndim(\n2\n)\n1,\n3\n1\n(\n0\n)\n(\n)\n,\n(\nh\nDiag\nt\nh\nC"},{"paragraph_id":"p287","order":287,"text":"), while \n)\ndim(\n)\ndim(\nh\nu \n.\nCorollary XII.1 :\nLet \n,0\n)\n)\n,\n(\n)\n,\n(\ndet(\n),\n2\n(\n)\ndim(\n,\n3\n)\ndim(\n)\ndim(\n)\n,...,\n1(\n)\n1,\n3\n1\n("},{"paragraph_id":"p288","order":288,"text":"t\nu\nC\nt\nh\nC\nn\nn\nt\nn\nh\nu\nn\nm\nM is an \nn\nn -matrix. Then\n)\n1\n1\n)\n...\n0\n)\n,\n(\n))\n,\n(\n(\n.........\n)\n,\n(\n))\n,\n(\n(\n)\n,\n(\n(,\n,\n,\n1\n(\n)\n1\n1(\n)\n(\n)\n},\n,...,\n1\n({\n)\n1\n},\n,...,\n1\n({\n2\n1\n)\n1\n,...,\n(\n)\n,...,\n1\n(\n1\n)\n1\n,...,\n(\n1\n)\n,...,\n1(\n)1,\n3\n1\n(\n3\n^"},{"paragraph_id":"p289","order":289,"text":"n\nn\nn\nn\nn\nn\nT\nn\nn\nn\nT\nn\nn\nM\nM\nt\nu\nC\nh\nu\nC\nt\nu\nC\nh\nu\nC\nt\nh\nC\nt\nt\nh\nu\ngen\nM\nper\nm"},{"paragraph_id":"p290","order":290,"text":"References:\n1) L.G. Valiant. The Complexity of Computing the Permanent.\nTheoret. Comp. Sci. 8, 189-201, 1979.\n2) Okada Soichi. Generalizations of Cauchy’s Determinant Identity \nand Schur’s Pfaffian Identity.\nhttp://citeseer.ist.psu.edu/cache/papers/cs2/125/http:zSzzSzwww.math.kobe-\nu.ac.jpzSzpublicationszSzrlm18zSz9.pdf/generalizations-of-cauchy-s.pdf\n3) Richard A. Brualdi, Herbert J. Ryser, Combinatorial Matrix Theory, \nCambridge University Press, Cambridge, 1991. Comm. Computer Algebra \n39 (2005), pp. 61–64.\n 4) Henryk Minc, Permanents, Addison-Wesley, Reading, 1978.\n 5) http://en.wikipedia.org/wiki/Pfaffian\n 6) http://en.wikipedia.org/wiki/Cauchy_determinant\n 7) Mark Jerrum. The computational complexity of counting. \nThe International Congress of Mathematicians, Zurich, August 1994.\nhttp://www.lfcs.inf.ed.ac.uk/reports/94/ECS-LFCS-94-296/index.html"}],"pages":[{"page":1,"text":"Vadim Tarin\n A Polynomial-time Algorithm for \nComputing the Permanent in GF( q\n3 ) \nAbstract: \nA polynomial-time algorithm for computing the permanent in any field of characteristic 3 is \npresented in this article. The principal objects utilized for that purpose are the Cauchy and \nVandermonde matrices, the discriminant function and their generalizations of various types. \nClassic theorems on the permanent such as the Binet-Minc identity and Borchadt's formula \nare widely applied, while a special new technique involving the notion of limit re-defined for \nfields of finite characteristics and corresponding computational methods was developed in \norder to deal with a number of polynomial-time reductions. All the constructions preserve a \nstrictly algebraic nature ignoring the structure of the basic field, while applying its infinite \nextensions for calculating limits.\nA natural corollary of the polynomial-time computability of the permanent in a field of a \ncharacteristic different from 2 is the non-uniform equality between the complexity classes \nP and NP what is equivalent to RP=NP( Ref. [1]).\n Unless specified otherwise, \n all the results are for fields of Characteristic 3.\nDefinitions and denotations.\nFor \nm\nn \n-matrices \nm\nn\nh\nij\ndef\nh\nm\nn\nij\nij\ndef\na\nA\nb\na\nB\nA\nB\nA\n \n \n \n \n \n \n \n}\n{\n,\n}\n{\n,\n,\n \n ,\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nq\nq\nh\nh\ndef\nh\nh\nA\nA\nA\n...\n1\n1\n)\n,...,\n(\n (for real numbers \nqh\nh\nh\n,...,\n,\n1\n)\n(the Hadamard product, degree and vector-degree),\nfor \n)\n,\n(\n}\n,...,\n1\n{\n},\n,...,\n1\n{\nJ\nI\nA\nm\nJ\nn\nI\n \n \n is the sub-matrix of A whose \nset of rows is I and of columns J ,"},{"page":2,"text":"for matrices \n]\n[\n]\n1[ ,...,\ns\nA\nA\n with the same number of rows \ns\nr\nr\nA\n1\n]\n[\n \n denotes \nthe matrix \n \n]\n[\n]\n1\n[\n|\n...\n|\ns\nA\nA\n,\n \n \n \n \n \nn\ni\nij\nm\nj\na\nA\nscal\n1\n1\n)\n(\n ,\n \n;\ns)\nfor vector\n(as\n)\n('\n),\n(\nsimply \n write\n will\nwe\n1\n)\ndim(\nfor\n,)\n(\n...\n)\n('\n,)\nmatrix\ne\nVandermond\nthe\n(\n)\n(\n)\n(\n,\n)\n(\n]\n[\n]\n[\n]\n[\n)\ndim(\n1\n]\n[\n)]\n[dim(\n)\n1\n,...,\n0\n(\n]\n[\nt\nvan\nt\nvan\nt\nt\nVan\nt\nd\nd\nt\nd\nd\nt\nVan\nt\nVan\nt\nVan\nt\nt\nVan\nh\nh\nh\nt\ndef\nh\nt\ndef\nh\nT\ndef\nh\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n2\n(mod\n1\n)\ndim(\n,\n)\n(\n)\n(\n),\n2\n(mod\n0\n)\ndim(\n,\n)\n(\n)\n(\n)\n(\n]\n2\n/)\n1\n)\n[(dim(\n]\n2\n/)1\n)\n[(dim(\n]\n2\n/)\n[dim(\n]\n2\n/)\n[dim(\nt\nt\nVan\nt\nVan\nt\nt\nVan\nt\nVan\nt\nW\nt\nt\nt\nt\ndef\n,\n)\n(\n)\n,\n(\n)\ndim(\n1\n)\ndim(\n1\n \n \n \n \n \n \nu\ni\nv\nj\nj\ni\ndef\nv\nu\nv\nu\npol\nfor a scalar \n)\n,\n(\n)\n,\n(\"\n,)\n,\n(\n)\n,\n('\n,\n2\n2\nv\npol\nd\nd\nv\npol\nv\npol\nd\nd\nv\npol\ndef\ndef\n \n \n \n \n \n \n \n \n \n \n \n ;\nfor an appropriate set I , \nIu\nis the sub-vector of u consisting of the coordinates \nwith indexes from I (including the case when I is a multi-set, with correspondingly \nduplicated coordinates),\nI\nu\\\nis its compliment in u ;\nfor a scalar , \nm\nn \n \n is the \nm\nn \n-matrix all whose entries equal ,\nfor \n1\n \nm\nwe will write simply \nn\n ;`\nfor a set H , \n)\n(H\nPart\n is the set of its partitions on nonempty sets;\nfor a finite sequence d, |d| denotes the number of its members;\nfor a real number \n \n \n \n \n \n0\n,1\n0\n,0\n)\n(\n,\nu\nu\nu\nu \n ;\nwhere \n)\ndim(\n|\n|,\n1,\n|\n|,\n1,\n)\ndim(\n)\ndim(\n)}\n,...,\n,\n,...,\n{(\n)}\n,\n{(\n}\n{\na\ni\ni\ni\ni\na\ni\ni\na\ni\ni\ni\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n is a \nmatrix)\nextension \nCauchy \nthe\n(\n}\n}\n1\n)!\n)(\n!\n(\n1\n)){\n(\n1\n{(\n)\n,\n(\n);\nmatrix\nCauchy \nthe\n(\n}\n1\n{\n)\n,\n(\n)\ndim(\n)\ndim(\n|\n||\n|\n,\n,\n)\ndim(\n)\ndim(\n,\n,\n,\n,\na\na\nj\ni\nj\ni\nl\nj\nk\ni\ndef\ny\nx\nj\ni\ndef\nj\ni\nl\nj\nl\nj\nk\ni\nk\ni\na\na\na\nd\nd\na\nd\nd\nj\ni\na\nE\ny\nx\ny\nx\nC"},{"page":3,"text":")\ndim(a -vector whose coordinates are pairs of non-decreasing nonnegative integer \nfinite sequences, including the empty one. The rows and columns containing the\nterm \nia will be called the \nia -extension (and \nia itself the extension’s value );\n)\n,\n(\ni\ni\ni\n \n \n \n we’ll call the extension-degree of \nia , where \ni is the left \nextension-degree, \ni\n is the right one, \n|\n|\ni \n is the extension’s height , \n|\n|\ni\n \nis its width , and \n|\n|\n|\n|\n)\n(\ni\ni\ndef\ni\nbal\n \n \n \n \n \n \n is its balance . If \n0\n|\n|\n|\n|\n \n \ni\ni\n \n \nthen we’ll say that the matrix has an \nia -singularity. Singularities with \n(0,0)\n \ni \nwill be called simple. The sum \n \n \n \n)\ndim(\n1\n)\n(\n)\n(\na\ni\ni\nbal\nbal\n \n \nwill be called the total balance\nof the extension-degrees’ vector .\n \nfield.\nbasic\n \nover the\nfunction \n-\nweight\nits\nis\n )\n(\ndegrees,\n-\nextension\nof\nset \na\nis\n \n),\ndim(\n1,...,\nfor \nwhere,\n))\n,\n(\n(\n))\n(\n(\n))\n(\n(\n...\n \nexpression\n the\nunderstand\nll\n we'\nsum\n-\nE\nan \nBy \n1\n1\n)\ndim(\n)\ndim(\n)\ndim(\n1\ni\ni\ni\na\ni\ni\ni\nweight\na\ni\na\nE\nper\nbal\nweight\na\na\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThe formal sum \n \n \n \n \ni\ni\ni\ni\ni\ni\nweight\n \n \n \n \n)\n(\n will be defined as the extension- plane\nof \nia , \n)\n(\ni\ni\nweight \n will also be called the \nweight\nplane -\n of \ni in \ni , the pair \n)\n,\n(\ni\nia \nwill be called an extension-variety and thus we can also talk about the E-sum of \na on the extension-planes \n),\ndim(\n1,...,\n \n,\na\ni\ni\n \n \n while the pair of vectors \n)\n,\n(\n \na\nwill be called a vector-variety. If \ni has the form \n)\n0,\n(\n)\n,\n(\n \n \n \n \ni \n then we’ll say \nthat the E-sum is right-prolonged on \nia , if \n)\n,0\n(\n)\n,\n(\n \n \n \n \ni \n then left-prolonged,\nand the corresponding patterns of \n)\n,\n(\n \na\n (i. e. sub-vectors of the vector-variety) will be \ncalled right and left prolongations (correspondingly).\nWe will denote the E-sum of a on by \n)\n,\n(\n \na\nesum\n (we’ll use the brackets\n \n only \nfor left and right extension-degrees whose cardinalities exceed 1);\n)\n)\n0,0\n(,\n(\n)\n(\n~\n),\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n)\n,\n,\n(\n~\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\nz\ndef\nz\ny\nx\ndef\nz\nE\nz\nC\nz\ny\nx\nE\nz\ny\nx\nC\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n ;\nfor an \nr\nn -matrix A and an\nr\nm -matrix B ,\nm\nn\nm\n,\n2\n \n is even :\n(the copermanent of A with B ) ,\n)\n(\n)\n(\n)\n,\n(\n})\n,\n{\n},\n,...,\n1\n({\n)\n},\n,...,\n1\n({\n2\n/\n|\n|,\n|\n|,\n:\n,\nJ\nJ\nm\nI\nn\nm\nJ\nn\nI\nJ\nI\nJ\nI\ndef\nB\nper\nA\nper\nB\nA\ncoper"},{"page":4,"text":",\n)\n(\n)\n(\n,\n)\ndet(\n)\ndet(\n|\n|,\n)\n},\n,...,\n1\n({\n|\n|,\n)\n},\n,...,\n1\n({\n \n \n \n \n \n \nn\nJ\nJ\nJ\nn\nn\nJ\nJ\nJ\nn\nA\nper\nA\nper\nA\nA\n \n \n \n \n \n \n \nm\nk\nJ\nj\nkj\nJ\nn\nn\nJ\nJ\ndef\nB\nb\nA\nper\nA\nr\ne\np\n1\n)\n},\n,...,\n1\n({\n|\n|,\n)\n(\n)\n(\nˆ\n(the permanent of A on the base B , or the base-permanent of A on B );\nfor r-vectors \n \n ,\n,u\n, nonnegative integers \nq\np \n:\n \n \n \n \n \n \n \n \n \n \nI\ni\nJ\nj\nj\ni\nJ\nI\nq\nJ\np\nI\nJ\nI\nJ\nI\ndef\nq\np\nu\nVan\nu\nVan\nu\ndis\n)\n)(\n))(\n(\n(\ndet\n))\n(\n(\ndet\n)\n,\n,\n(\n2\n2\n|\n|,\n|\n|,\n:\n,\n,\n \n \n \n \n(the \nq\np, -discriminantal of u on \n \n ,\n) ;\nthe E-generated functions (for vectors with appropriate dimensions):\nthe star-function:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n2\n/)\ndim(\n|\n|,\n,\n,\n}\n,\n,\n{\n\\\n]\n2\n[\n]\n1\n[\n]\n2\n[\n]\n1\n[\n))\n(\n)\n,\n(\n(\n))\n,\n(\n(\n)\n)(\n(\n)\n,\n,\n,\n,\n(\nz\nJ\nI\nJ\nI\nJ\nI\nI\ni\nJ\nJ\nI\nJ\nI\nJ\nI\nJ\nj\nj\ni\ndef\ndef\nDiag\nu\nz\nC\nper\nz\nz\nC\nper\nu\nz\ngen\n \n \n \n \n \n \nthe 2-waves-function:\n)\n)\n0,0\n(\n,\n(\n)\n)\n0,0\n(\n)\n0,\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n)\n,\n,\n,\n,\n,\n(\n|\n|\n)\ndim(\n\\\n\\\n\\\n|\n|\n|\n|\n)\ndim(\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nw\nI\nJ\nI\nI\nJ\nJ\nI\ny\nx\nJ\nI\ndef\ndef\nw\nz\nesum\nw\nz\nt\nh\nesum\nt\nh\nw\nz\ngen\n \n \n \n \n)\n0,1(\n)1,0\n(\n)\n0,0\n,\n0,0\n(\n \n where\n \n \n \n \n \n \n \n \n \n \n \n \n \n \ndef\n.\nWe’ll call the extension-plane (0,0) the wave and the biwave; together with \ntheir values (say \niz and \nj\nw ) such extension-varieties will be called a \niz -wave \nand a \nj\nw -biwave (correspondingly) ; \nthe wave-function (a partial case of the 2-waves-function):\n;\n))\n(\n~\n(\n)\n))(\n,\n,\n(\n~\n(\n)\n,\n,\n,\n,\n,\n(\n)\n,\n,\n,\n,\n(\n)\ndim(\n|\n|,\n\\\n~\n \n \n \n \n \n \n \n \n \n \n \nI\nh\nJ\nJ\nJ\nj\nj\nI\nI\ni\ni\nI\nJ\ndef\nz\nC\nper\nz\nt\nh\nC\nper\nt\nh\nz\ngen\nt\nh\nz\ngen\n \n \n \n \n \n \nFor both the wave-function and the 2-waves-function \ni\n will be called the i -th ebb, \nand the ebb vector.\nthe base-function:"},{"page":5,"text":"I\nt\nu\nC\nI\ni\nh\ni\nh\nh\nt\nh\nh\nt\nu\nC\nh\nu\nC\ndef\nDiag\nt\nh\nC\nr\ne\np\nh\nu\nC\nDiag\nt\nh\nC\nI\nr\ne\np\nt\nh\nu\ngen\nI\n))\n(\n)\n,\n(\n(\nˆ\n))\n1\n)\n,\n(\n(\n(\n)\n)\n(\n)\n,\n(\n0\n0\n(\nˆ\n)\n,\n,\n,\n(\n)\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\n,\n(\n)\n,\n(\n^\n\\\n \n \n \n \n \n \nif F is a basic field and is a formal variable then for\n \n \n \n \n \n \n \n \n \n \n \n \n0\n)\n(\n,\nexist\nt \ndoesn'\n0\n)\n(\n,\n)\n(\nlim\n,\n)\n(\n:)\n(\n]\n0\n[\n0\n]\n[\ng\norder\ng\norder\ng\ng\nq\ng\norder\nF\ng\ng\nq\nk\nk\nk\n \n \n \n \n \n \n \nBy \n)\n),\n(\n(\nx\nx\nf\n \n we’ll denote the Jacobian matrix \n)\ndim(\n)\ndim(\n}\n)\n(\n{\nx\nf\nj\ni\nx\nx\nf\n \n \n \n .\nPreliminaries.\nThe Binet-Minc identity (for any characteristic, Ref. [3], [4]):\nLet A be an \n \n m\nn\nmatrix, then\n)\n(\n})\n,...,\n1\n({\n1\n)!\n1\n|\n(|\n)1\n(\n)\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \nn\nPart\nI\nm\nj\nI\ni\nij\nn\na\nI\nA\nper\n,\nThe Binet-Minc identity for Characteristic 3:\nLet A be an \n \n r\nn\nmatrix, then\n \n \n \n \n \n \n \n \n \n \n})\n,...,\n1\n({\n)\n,\n(\n3\n,\n2\n,1\n)\n(\n)\n(\n)!\n1\n|\n(|\n)1\n(\n)\n(\nn\nPart\nI\nR\nI\nn\nA\nscal\nI\nA\nper\n ,\nwhere:\n)\n(\n3,2\n,1\nH\nPart\n is the set of partitions of the set H on nonempty sets of cardinalities \nnot bigger than 3, \n}\n,...,\n1\n{\nr\nR \n.\nFor our purposes we’ll also use this identity in the form \n \n \n \n \n \n \n \n \n \n \n})\n,...,\n1\n({\n)\n,\n(\n|\n|\n1\n3\n,\n2\n,1\n)\n(\n)\n(\n)!\n1\n|\n(|\n)1\n(\n)\n(\nn\nPart\nI\nR\nI\nI\nA\nscal\nI\nA\nper\nThe generalized Binet-Minc identity for \n)\n,\n(\nB\nA\ncoper\n :\nLet A , B be an \nr\nn - and \nr\nm -matrix, \n)\n2\n(mod\n0\n \nm\n, then\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nK\nn\nN\nK\nm\nA\nscal\nI\nB\nA\nscal\nB\nA\ncoper\n)\n(\n)\n(\n)\n(\n)\n)!\n1\n|\n(|\n(\n)\n(\n)1\n(\n)1\n(\n)\n,\n(\n)\n,\n(\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|\n|\n2\n/\n3\n,\n2\n,1\n2\n;\n2\n,1"},{"page":6,"text":"where:\n},\n,...,\n1\n{\n},\n,...,\n1\n{\n},\n,...,\n1\n{\nr\nR\nm\nM\nn\nN\n \n \n \nfor two disjoint sets \n'\nH and \n\"\nH ,\n)\"\n,'\n(\n2\n;\n2\n,1\nH\nH\nPart\n is the set of partitions of the set \n\"\n' H\nH \non sets having 1 or 2 elements in \n'\nH and 2 elements in \n\"\nH (those sets are denoted here as \npairs \n\"\n\"\n,'\n'\n),\n\"\n,'\n(\nH\nI\nH\nI\nI\nI\n \n \n).\nProof:\nthe right part can be re-written as \n)\n(\n)\n(\n)\n(\n)1\n(\n)!\n1\n|\n(|\n)\n(\n)\n,\n(\n1\n|\n|\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n3\n,\n2\n,1\n2\n;\n2\n,1\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nN\nK\nA\nscal\nI\nB\nA\nscal\n. \nLet’s denote it by \n)\n,\n(\nB\nA\n \n.\nThen, by the construction,\n)\n(\n}\n,...,\n{\n)\n(\n}\n,...,\n{\n}\n,...,\n1\n{\n,\n,\n'\n\"\n\"\n'\n|\n|\n|\n|\n\\\n\"\n|\n|\n\"\n1\n'\n|\n|\n'\n1\n\"\n1\n'\n1\n\"\n|\n|\n'\n|\n|\n'\n\"\n\"\n'\n)\n1,\n1\n(\n...\n)\n,\n(\nM\nPart\nI\nI\nN\nPart\nI\nI\nr\nJ\nI\nI\nI\nI\nJ\nj\nI\ni\nI\ni\nj\ni\nj\ni\nI\nI\nJ\nJ\nJ\nJ\nj\nj\nj\nj\nb\na\nB\nA\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere \n \n\\}\n{\n means the set of sets with removed -s.\nThen let's prove that \n))\n2\n(\n)\n(\n)(\n1\n(\n)\n1,\n1\n(\n \n \n \n \ns\ns\nq\ns\nq\n \n \n \n \n \n \n .\nThe generalized Binet-Minc identity for \n)\n(\nˆ\nA\nB\nr\ne\np\n:\nLet A , B be an \nr\nn - and \nr\nm -matrix, then\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nI\nI\nK\nn\nN\nK\nB\nA\nscal\nI\nB\nA\nscal\nA\nr\ne\np\n)\n(\n)\n(\n)\n(\n)\n)!\n1\n|\n(|\n(\n)\n(\n)1\n(\n)1\n(\n)\n(\nˆ\n)\n,\n(\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|)\n\"\n|\n1\n)(\n2\n|'\n(|\n|\n|\n3\n,\n2\n,1\n1\n;\n2\n,1\n)\n(\n \nwhere:\n},\n,...,\n1\n{\n},\n,...,\n1\n{\n},\n,...,\n1\n{\nr\nR\nm\nM\nn\nN\n \n \n \nfor two disjoint sets \n'\nH and \n\"\nH ,\n)\"\n,'\n(\n1\n;2\n,1\nH\nH\nPart\n \n is the set of partitions of the set \n\"\n' H\nH \non sets having 1 or 2 elements in \n'\nH and at least 1 element in \n\"\nH (those sets are denoted \nhere as pairs \n\"\n\"\n,'\n'\n),\n\"\n,'\n(\nH\nI\nH\nI\nI\nI\n \n \n).\nProof:\nby analogy with the previous proof, we re-write the right part as\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nR\nI\nI\nK\nN\nPart\nM\nK\nPart\nI\nI\nR\nI\nR\nI\nI\nI\nN\nK\nA\nscal\nI\nB\nA\nscal\n)\n(\n)\n(\n)\n(\n)1\n(\n)!\n1\n|\n(|\n)\n(\n)1\n(\n)\n,\n(\n1\n|\n|\n)\n\\\n(\n),\n,\n(\nˆ\nˆ\n)\"\n,'\n(\n)\n,\"\n(\n)\n,'\n(\n|)\n\"\n|\n1\n)(\n2\n|'\n(|\n)\n(\n3\n,\n2\n,1\n1\n;\n2\n,1\n \nLet’s denote it by \n)\n,\n(ˆ\nB\nA\n \n."},{"page":7,"text":"Then, by the construction,\n)\n(\n}\n,...,\n{\n)\n(\n}\n,...,\n{\n}\n,...,\n1\n{\n,\n,\n'\n\"\n\"\n'\n|\n|\n|\n|\n\\\n\"\n|\n|\n\"\n1\n'\n|\n|\n'\n1\n\"\n1\n'\n1\n\"\n|\n|\n'\n|\n|\n'\n\"\n\"\n'\n)\n1,\n1\n(ˆ\n...\n)\n,\n(ˆ\nM\nPart\nI\nI\nN\nPart\nI\nI\nr\nJ\nI\nI\nI\nI\nJ\nj\nI\ni\nI\ni\nj\ni\nj\ni\nI\nI\nJ\nJ\nJ\nJ\nj\nj\nj\nj\nb\na\nB\nA\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThen let's prove that \n)1\n(\n)\n1,\n1\n(ˆ\n \n \nq\ns\nq\n \n \n \n \nLemma 1 (known, for any characteristic, Ref. [2], [6]):\nLet \n).\ndim(\n)\ndim(\ny\nx \n Then\n)\n,\n(\n))\n(\ndet(\n))\n(\ndet(\n)1\n(\n))\n,\n(\ndet(\n2\n)\n1\n)\n)(dim(\ndim(\ny\nx\npol\ny\nVan\nx\nVan\ny\nx\nC\nx\nx\n \n \n \nCorollary 1.1 :\nLet \n).\ndim(\n)\ndim(\n2\ny\nx \n Then\n)\n,\n(\n))\n(\ndet(\n))\n(\n(\ndet\n)\n)\n,\n(\n)\n,\n(\ndet(\n2\n4\n2\ny\nx\npol\ny\nVan\nx\nVan\ny\nx\nC\ny\nx\nC\n \n \n \n \n \n \n \n \nProof:\n)\n(\n)\n(\n))\n,\n1\n(\ndet(\n)1\n(\nlim\n))\n)\n,\n1\n(\n)\n,\n(\ndet(\nlim\n)\n)\n,\n(\n)\n,\n(\ndet(\n)\ndim(\n)\ndim(\n)\ndim(\n0\n)\ndim(\n)\ndim(\n0\n2\ny\nx\nx\nC\ny\nx\nC\ny\nx\nC\ny\nx\nC\ny\nx\nC\nx\nx\nx\nx\nx\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nLemma 2 (for any characteristic, Ref. [5]):\nLet A , B be \nm\nn \n-matrices, \nm\nn \n. Then \n(*) \nFor \n)\n2\n(mod\n0\n \nn\n: \n)\n)}\n(\n{\ndet(\n)\n(\ndet2\nT\nm\nm\nA\nj\ni\nsign\nA\nA\n \n \n \n, \n)\n)}\n(\n({\n)\n)}\n(\n{\n(\n)\ndet(\nn\nn\nT\nm\nm\nj\ni\nsign\nPf\nA\nj\ni\nsign\nA\nPf\nA\n \n \n \n \n \nwhere for a real \n \n \n \n \n \n \n \n \n0\n,0\n0\n,1\n0\n,1\n)\n(\nu\nu\nu\nu\nsign\nu"},{"page":8,"text":"For \n,\n,...,\n1\n,\nm\nq\nn\n \n \nscalar :\n)\ndet(\n)\n0\n0\n0\n2\n1\n1\n0\n0\n0\ndet(\n2\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n2\n2\n)\n2\n(\n2\nA\nI\nI\nA\nq\nm\nq\nm\nq\nq\nm\nq\nm\nq\nq\nm\nq\nq\nq\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n. \nIn the generic case this method allows to reduce A to a triangular matrix\nwhen computing \n)\ndet(A .\n(**) for \n)\n2\n(mod\n0\n \nn\n \n \n \n )\n|\n|\ndet(\n)\n|\ndet(\n1\n)\n,\n(\n)\n,\n(\n)\n,\n(\n)\n,\n(\n2\n/\n|\n|,\n)\n,\n(\n)\n,\n(\n \n \nm\nk\nk\nN\nk\nN\nk\nN\nk\nN\nn\nJ\nJ\nJ\nj\nj\nN\nj\nN\nB\nA\nB\nA\nB\nA\n \n \n \n \n \n \n \n ,\nwhere \n}\n,...,\n1\n{\nn\nN \n.\nLemma 3 (for any characteristic):\nLet \n)\n(x\nf\n be a polynomial in \n)\ndim(\n1,...,\nx\nx\nx\n of degree d . Then \n)))\n(\n(\n(\n)\n(\n0\n \n \n \n \n \n \n \n \n \nx\nx\nf\ncoef\nf\nd\ni\ni\nwhere \ni\ncoef is the coefficient at \ni (if the expression is considered as a polynomial in ).\nThe neighbouring computation principle (for any characteristic):\nLet \n)\n(u\nf\n be a polynomial in \n)\ndim(\n1,...,\nu\nu\nu\n of degree d over the field F , \n)\n,...,\n(\n)\ndim(\n1\nh\nh\nh\nu\nu \n , u exists at \n]\n0\n[h\n;\n)\n,...,\n(\n)\ndim(\n1\nh\nh\nh\nv\nv \n, \n)\ndim(\n]\n0\n[\n0\n)\n(\nv\nh\nv\n \n \n;\n)\n,\n(\n]\n0\n[h\nv\nu\n \n \n \n \n \n \n \n exists and is nonsingular. \nThen, given , over \n)\n( \nF\n: \n)))\n(\n(\n(\n)\n(\n0\n]\n[\n0\n \n \n \n \n \nd\nk\nk\nk\nd\ni\nh\nu\nf\ncoef\nf\ni\n \n \n \n.\nwhere:\nd\nk\nh\nu\ncoef\nh\nu\nk\nh\nh\nv\nu\nv\nk\ni\ni\ni\nk\nk\n,...,\n1\n,\n0\n))\n(\n(\n)\n)\n(\n)(\n1\n(\n)\n,\n(\n)\ndim(\n1\n0\n]\n[\n]\n0\n[\n]\n[\n]\n0\n[\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(And thus \n)\n(\n)\n)\n(\n(\n)\n(\n)\n(\n1\n]\n0\n[\n]\n0\n[\n0\n]\n[\n \n \n \n \n \n \n \n \nd\nd\nk\nk\nk\nO\nh\nu\nh\nu\nh\nu\n \n \n \n \n )\nThis method will be called the \nng\nneighbouri\n computation of the function \n)\n( \nf\n by the\nzation\nparameteri\n)\n(h\nu\n in the region\n)\ndim(\n0\n)\n(\nv\nh\nv\n \n \n from the \nbearing point \n]\n0\n[h\n."},{"page":9,"text":"If \n))\n(\n(\nh\nu\nf\n is computable in polynomial time for any h in the region\n)\ndim(\n0\n)\n(\nv\nh\nv\n \n \n and there exists a bearing point, then \n)\n( \nf\n is computable \nin polynomial time for any too. \nIn the further, to be clear and short, we’ll call a system of functions S \nalgebraically absolutely independent in a region R (given by a system \nof equations with a zero right part) iff the joint system of functions consisting \nof S and the left part of the system representing R is algebraically independent \nat some point of R. \nThe prolongation-derivative principle (for any characteristic):\na left (right) prolongation pattern of a vector-variety can be removed with an \ninduced coordinate-wise transformation of the remaining vector of extension-planes \nsuch that the E-sum preserves its value. Such a transformation will be called a left (right) \nprolongation-derivative, or the prolongation-derivative on the given prolongation. \nFormally it will be denoted (correspondingly for the left and right cases) by\n)\n)1,\n(\n)\n,\n(\n,\n(\n)\n)\n,\n,\n(\n,\n(\n)\n,\n(\n)\n,\n(\n)\n)\n,1(\n)\n,\n(\n,\n(\n)\n)\n,\n,\n(\n,\n(\n)\n,\n(\n)\n,\n(\n)\ndim(\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nesum\nd\nd\nesum\nesum\nd\nd\nesum\nd\nd\nesum\nesum\nd\nd\ndef\nR\nR\ndef\nR\nR\ndef\nL\nL\ndef\nL\nL\nThis principle is based on the Binet-Minc identity. \n \nLemma 4:\nLet A be an \nn\nn \n-matrix, B be a \nn\nn \n)\n2\n(\n-matrix. Then \n \n \n \n \n)\n(\ndet\n)\n|\n(\n)\n|\n(\n2 A\nB\nB\nper\nBA\nBA\nper\n \nThe Borchardt formula (for any characteristic, Ref. [2]) :\nLet \n)\ndim(\n)\ndim(\ny\nx \n. Then\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n))\n,\n(\n(\n2\ny\nx\nC\ny\nx\nC\ny\nx\nC\nper\n \n \nLemma 5:\n))\n(\ndet(\n)\ndet(\n))\n(\n(\n)\ndim(\nt\nVan\nt\nt\nW\nper\nt\n \n \n \nwhere \n)\ndim(t\n \nis the first \n)\ndim(t members of the sequence \ncomposed as the sequence of pairs \n \n \n \n,...,\n0\n),\n1\n3,\n3\n(\ns\ns\ns"},{"page":10,"text":"Proof:\n \n \n \n \n \n \n)\ndim(\n1\n0\n))\n(\n(\n))\n,\n(\n(\nlim\n))\n(\n(\ny\nj\njt\nx\nW\nper\nt\nx\nC\nper\nt\nW\nper\n \n \n \nThe main part.\nTheorem I :\nLet \nn\nu\nt\n2\n)\ndim(\n2\n)\ndim(\n \n \n.\nThen\nProof:\n))\n(\n(\ndet\n)\n,\n(\n))\n(\n(\n)\n,\n(\n)\n)\n(\n0\n0\n)\n(\n)\n(\n(\n)\n,\n(\n))\n,\n(\n)\n)}\n,\n(\n({\n(\n))\n,\n(\n(\n2\n2\n1\n2\nu\nR\nu\nt\npol\nt\nW\nper\nu\nt\npol\nu\nR\nu\nR\nt\nW\nper\nu\nt\npol\nu\nt\nC\nu\nt\npol\nDiag\nper\nu\nt\nC\nper\nT\nT\nn\ni\ni\nn\ni\n \n \n \n \n \n \n \n \n \n \n \n \nwhere \n)\n(u\nR\n is the matrix of the coefficients of the polynomials\nin the formal scalar variable \n)\n,\n(\n),...,\n,\n(\n\\\n1\\\nn\nu\npol\nu\npol\n \n \n \n. The determinant \nof this matrix is a homogenous polynomial in \nn\nu\nu ,...,\n1\n of degree \n2\n)1\n(\n)1\n(\n...\n1\n0\n \n \n \n \n \n \nn\nn\nn\n, and in the meantime it is divided by \nj\ni\nu\nu \nfor any \nn\nj\ni\n \n \n \n1\n, or by \n))\n(\ndet(\nu\nVan\n; moreover, its coefficients \nare 1 and -1, therefore \n))\n(\n(\ndet\n))\n(\n(\ndet\n2\n2\nu\nVan\nu\nR\n \n, what completes \nthe proof.\nCorollary I.1 :\nLet \n)\ndim(\n)\ndim(\n2\n)\ndim(\nv\nu\nt\n \n \n.Then\n)\n,\n(\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n))\n,\n(\n(\n2\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nu\nt\npol\nv\nu\npol\nu\nVan\nv\nt\nW\nper\nv\nu\nu\nt\nC\nper\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n))\n,\n(\n(\n2\nu\nt\npol\nu\nVan\nt\nW\nper\nu\nu\nt\nC\nper"},{"page":11,"text":"Proof:\n))\n,\n1\n(\n(\nlim\n))\n,\n(\n(\n)\ndim(\n)\ndim(\n0\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nv\nu\nu\nv\nt\nC\nper\nv\nu\nu\nt\nC\nper\nv\nv\n \n \n \n \nand, basing on Theorem I, we calculate the limit to get the lemma’s equality.\nCorollary I.2 :\nLet \n)\ndim(\n)\ndim(\n2\n)\ndim(\nv\nu\nt\n \n \n+1. Then\nProof:\n)\n/\n))\n/\n1\n,\n(\n(\n(\nlim\n)\n1\n)\n,\n(\n(\n0\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nu\nu\nt\nC\nper\nv\nu\nu\nt\nC\nper\nt\n \nand then, basing on Corollary I.1, we again just calculate the limit.\nTheorem II :\n)\n)\n(\n)\n(\n0\n)\n('\n)\n(\n)\n('\n)\n(\ndet(\n)\n,\n,\n(\n)\ndim(\n1\n]\n[\n]\n[\n]\n[\n]\n[\n]\n[\n]\n[\n,\n \n \nu\ni\ni\nq\np\ni\nq\np\nq\np\ni\nq\np\ni\ni\nq\np\ni\ni\nq\np\ni\ni\nq\np\ni\nq\np\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\nvan\nu\ndis\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nProof:\nbasing on Lemma 2, we conclude that the right part of this equality is\n \n \n \n \n \n \n \n \n \n \n \n \n \n \np\nI\nI\nI\ni\ni\nq\np\nq\np\ni\nq\np\ni\ni\nq\np\ni\nu\nvan\nu\nvan\nu\nvan\n|\n|,\n]\n[\n]\n[\n]\n[\n)\n)\n(\n0\n)\n('\n)\n(\ndet( \n \n \n \nTheorem III :\nLet\n)\n,\n(\n)\n,\n('\n,0\n)\n,\n(\"\n,1\n3\n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n|\n|\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \ny\nx\npol\nt\npol\ny\nx\npol\nn\nt\nn\ny\nn\nx\nI\n \n \n \nThen\n)\n,\n(\n)\n,\n(\n))\n(\n(\ndet\n))\n(\n(\n)\n1\n)\n,\n(\n(\n2\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nv\nu\nt\npol\nv\nu\npol\nu\nVan\nv\nt\nW\nper\nv\nu\nu\nt\nC\nper\nt"},{"page":12,"text":"I\ni\ni\nI\nI\ny\ny\npol\ny\nVan\nt\nx\npol\nx\nt\nW\nper\ny\nx\nC\nper\n)\n,\n('\n1\n))\n(\n(\ndet\n)\n,\n(\n))\n(\n(\n))\n,\n(\n(\n2\nProof:\nLet us apply the Binet-Minc identity to \n)\n)\n1\n|\n)\n,\n(\n((\n)\ndim(\nT\nt\nI\nI\ny\ny\nx\nt\nC\nper\n \n \n \n \n \n \n \n \n \n \n \nTheorem IV :\nLet\n)\n,\n(\n)\n,\n('\n,0\n)\n,\n(\"\n)\n2\n(mod\n0\n)\ndim(\n,1\n3\n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \ny\nx\npol\nt\npol\ny\nx\npol\nm\nz\nn\nt\nn\ny\nn\nx\n \n \n \nThen\n \n))\n(\n)\n,\n(\n),\n(\n)\n,\n(\n(\n \n \nDiag\ny\nz\nC\nDiag\ny\nx\nC\ncoper\n)\n}\n)\n,\n(\n{,\n}\n)\n,\n('\n{,\n(\n)\n,\n(\n))\n(\n(\n))\n(\n(\n2\n2\n2\n2\n/\n,\nn\ni\ni\nn\ni\ni\nm\nn\nz\ny\npol\ny\ny\npol\ny\ndis\nt\nx\npol\nz\nW\nper\nx\nt\nW\nper\n \n \n \n \n \n \n \n \n \nProof:\nIt follows from the definition of the copermanent and Theorems I, III. \n----------------------------------------------------------------------------------------\nLet’s note that for \n0\n)\n,\n(\"\n \n \n \n \n \n \n \ny\nx\npol \n to be true it’s sufficient that \n0\n)\n,\n(\"\n \n \n \n \n \n \n \ny\nx\npol \n for any n pair-wise distinct values of (due to the characteristic), \nfor instance for all the coordinates of x , i.e. it’s equivalent to \nn\nn\ni\ny\nx\nx\npol\n0\n)}\n,\n(\"\n{\n \n \n \n \n \n \n \n,\nor \nn\nn\nn\nn\ni\ni\ny\nx\nC\nx\nC\ny\nx\nx\npol\ny\nx\nx\npol\n0\n1\n)\n,\n(\n1\n)\n(\n~\n)}\n,\n('\n/)\n,\n(\"\n{\n2 \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n.\nTheorem V :\nLet\n),\n2\n(mod\n0\n)\ndim(\n,\n)\ndim(\n \n \n \nm\nz\nn\nx\n3,2\n,\n1\n)\n,\n(\n \n \n \nh\ny\nx\nC\nn\nh\nh\n \n \n \n. \nThen"},{"page":13,"text":"n\ns\ns\nm\nz\nz\ny\nx\nC\nDiag\ny\nz\nC\nDiag\ny\nx\nC\ncoper\ny\nz\nC\ny\nz\nC\ny\nx\nC\nx\nz\ngen\n1\n)\n2\n/\n1\n(\n2\n/\n)\ndim(\n3\n3\n)\ndim(\n2\n2\n)\n1\n(\n*\n)\n,\n(\n))\n(\n)\n,\n(\n),\n(\n)\n,\n(\n(\n)1\n(\n)\n1\n)\n,\n(\n,\n1\n)\n,\n(\n,\n)\n)\n,\n(\n(,\n,\n(\n \n \n \n \n \n \n \n \n \n \nProof:\nIt follows from the Binet-Minc identity for \n)\n,\n(\nB\nA\ncoper\nTheorem VI :\n Let\nn\nm\nz\nn\ny\nn\nx\n \n \n \n \n)\ndim(\n,\n2\n)\ndim(\n,\n)\ndim(\n. \nThen there exist scalar constants \n3\n2, \n \n such that the system of functions in \n \n,\n, y\nx\n($) \n \n \n \n \n \n \n \n \n \n \n \n3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n3,2\n,\n)\n,\n(\n:\n,..,\n1\n)\n(\nq\ns\nq\ny\nx\nC\nx\nz\nC\ns\ny\nz\nC\nm\nk\ns\nk\nq\ns\nk\n \n \nis algebraically absolutely independent in the region \n)\n( \n \n \n \n \n \n \n \n \n \n3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nn\ns\ns\nn\nn\nn\n \n \n \n \n \nProof:\nlet us consider the Jacobian matrix which is\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n,\n)}\n,\n(\"\n{\n)\n)\n,\n(\n)(\n,\n(\n)\n)\n,\n(\n)(\n,\n(\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n)\n,\n(\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n)\n2\n,1(\n)\n1\n(\n3\n2\n \n \n \n \n \n \ny\nx\ny\nx\nx\npol\ny\nx\nC\nx\nz\nC\ny\nx\nC\nx\nz\nC\ny\nx\nC\nx\nz\nC\ny\nz\nx\nC\ny\nz\nx\nC\nn\ni"},{"page":14,"text":")\n2\n(\n2\n2\n2\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n3\n(\n3\n)\n3\n(\n2\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n)\n2\n(\n)\n(\n3\n2\n)\n2\n(\n3\n2\n0\n)\n,\n(\n)\n1\n)\n,\n(\n1\n)\n(\n~\n(\n)\n(\n~\n0\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)\n(\n)\n,\n(\n0\n)\n)\n,\n(\n(\n)\n(\n)\n,\n(\nn\nn\nn\nn\nn\nm\nn\nm\nn\nn\nm\nn\nm\ny\nx\nC\ny\nx\nC\nx\nC\nDiag\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\ny\nx\nC\nDiag\nDiag\ny\nz\nx\nC\n \n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere -s denote some matrices of appropriate dimensions. For the purpose of \nsimplicity, let’s permute in a certain way the Jacobian’s rows and multiply some \nof them by -1, thus reducing it to the form \n \n \n \n \n \n \n \n \n22\n21\n)\n2\n(\n)\n2\n3\n(\n11\n0\nA\nA\nA\nn\nn\nm\n \n where\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n22\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\nA\n \n \n \nIf both the matrices are nonsingular then the Jacobian matrix is nonsingular too. \nIf F is the basic field, let’s consider them over \n)\n( \nF\n for \n)).\ndim(\n(mod\n0\n)\ndim(\n,\nover \n \n vectors\nare\n )\n(\n,\n,\n,\n \nwhere\n,\n,\n1\n,\n,\n1\n,\n]\n[\n]\n[\n]\n1\n[\n0\n]\n[\n)\ndim(\n/)\ndim(\n2\n]\n0\n[\n0\n]\n[\n)\ndim(\n/)\ndim(\n]\n0\n[\n]\n1[\n]\n0\n[\nx\nx\nF\nk\ny\nx\nh\nh\ny\ny\ny\nh\nx\nx\nx\nx\nk\nk\nk\nk\nk\nh\nx\nk\nk\nk\nh\nx\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFirst we’ll show how an -point (satisfying the region’s conditions \n)\n( )\nof such a type is built and then we’ll prove that in the generic case it gives \nnonsingular \n11\nA ,\n22\nA .\nIn order to build such an -point, it’s sufficient that the Jacobian matrix of \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n(\n)\n,\n(\n0\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n1\n)\n,\n(\n1\n)\n(\n~\n(\n)\n(\n~\n)\n(\n)\n,\n(\n)\n)\n,\n(\n(\n3\n2\n)\n2\n(\n)\n2\n(\n)\n2\n(\n)\n3\n(\n)\n3,2\n(\n2\n2\n2\n2\n2\n3\n2\n3\n2\n11\n \n \n \n \nDiag\ny\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nx\nC\nDiag\nx\nC\nDiag\ny\nx\nC\ny\nx\nC\nDiag\nA\nn\nm\nn\nm\nn\nn"},{"page":15,"text":"the system of functions representing the left part of \n)\n( exists and is nonsingular\nat a certain approximation of the power series (in ) for\n \n,y\nwhich satisfies\nthe region’s conditions up to some degree of (then all the other members of \nthe power series are computable via corresponding linear equations involving \nthe Jacobian matrix calculated for the approximation). Let’s consider only the first \ntwo members of the power series for y (i.e. \n]\n0\n[y\n+\n \n]\n1\n[y\n) and the first member of the \none for (i.e. \n]\n0\n[ \n).\nFor \n11\nA , let’s multiply its first two block-rows by \n2\n and then consider its\n0\nlim\n \n \n. In this limit let’s substitute the matrix received via summing up all \nthe columns corresponding to the same \nrh in \n)\ndim(\n,...,\n1\n,\n]\n0\n[\nh\nr\ny\n \n, for its \nfirst block-column (such a transformation can’t enlarge the rank) .\nAnd now \n22\nA :\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n(\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n22\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nC\nDiag\ny\nx\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nx\nC\nA\n \n \n \n \n \n \n \nP\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\nx\nz\nC\nDiag\ny\nz\nC\ny\nx\nC\nDiag\ny\nx\nC\nP\nI\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\ny\nx\nC\nDiag\nx\nz\nC\ny\nx\nC\nDiag\ny\nz\nC\ny\nx\nC\nDiag\ny\nx\nC\nP\nDiag\ny\nx\nC\nn\nm\nn\nm\nn\nm\ni\nj\nj\nn\nn\nm\nn\nn\nn\nn\nn\nm\ni\nj\nj\nn\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)\n)\n)\n,\n(\n((\n)\n,\n(\n0\n)}\n)\n(\n)\n,\n(\n)\n,\n(\n)\n(\n)\n,\n(\ndet(\n)\ndet(\n)1\n(\n{\n0\n0\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)\n)\n)\n,\n(\n((\n)\n,\n(\n)\n,\n(\n)}\n)\n(\n)\n,\n(\n)\n,\n(\n)\n(\n)\n,\n(\ndet(\n)\ndet(\n)1\n(\n{\n)\n(\n)\n,\n(\n)\n3\n(\n)\n2\n,1\n(\n)\n2\n(\n\\\n)\n3\n(\n)\n2\n,1(\n)\n2\n(\n\\\n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere \n \n \n \n \n \n \n \n)\n,\n(\n)\n(\n)\n,\n(\ny\nx\nC\nDiag\ny\nx\nC\nP\n \n .\nSince we use the expression \n)\ndet(\n1\nP , first let’s prove the non-singularity of P\nin the generic case. It’s clearly seen for \n)\ndim(\n)\ndim(\nx\nh \n. \nCorollary VI.1 :\n)\n,\n,\n,\n,\n(\n]\n2\n[\n]\n1[\n \n \n \nu\nz\ngen \n (which, according to the Binet-Minc identity, \nis a function in the essential variables: the Cauchy - Binet - Minc weights\n)\n3\n,...,\n,2,1\n(\n)\n,\n(\nq\ns\nq\nu\nz\nC\ns\ni\nq\n \n \n \n \n \n and the star - functional weights\n]\n2\n[\n]\n1\n[ ,\ni\ni\n \n \n)"},{"page":16,"text":"is computable in polynomial time for any values of the essential (and, hence, \noriginal) variables via a neighbouring computation based on Theorems V, VI \nas the following:\nthe parameterization (of the essential variables) in \n \n,\n, y\nx\n:\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n2,1\n,\n1\n)\n,\n(\n)\n2\n(mod\n0\n)\ndim(\n)\n(\n1\n1\n]\n[\nq\ns\nq\ny\nx\nC\nx\nz\nC\nu\nz\nC\ns\ny\nz\nC\nm\nz\ns\ns\nq\nm\ns\ns\ns\n \n \n \n \n \n \nthe region : \n \n \n \n \n \n \n \n \n \n \n \n3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\n)\ndim(\n2\n)\ndim(\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nx\ny\nn\nn\ns\ns\nn\nn\nn\n \n \n \n \n \nHence the final formula for \n)\n,\n,\n,\n,\n(\n]\n2\n[\n]\n1[\n \n \n \nu\nz\ngen \n is:\n)\n}\n)\n,\n(\n{,\n}\n)\n,\n('\n{,\n(\n)\n,\n(\n))\n(\n(\n))\n(\n(\n)\n,\n,\n,\n,\n(\n2\n2\n2\n/\n,\n]\n2\n[\n]\n1[\nn\ni\ni\nn\ni\ni\nm\nn\nz\ny\npol\ny\ny\npol\ny\ndis\nt\nx\npol\nz\nW\nper\nx\nt\nW\nper\nu\nz\ngen\n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere: \n)\n2\n(mod\n0\n)\ndim(\n,\n2\n)\ndim(\n2\n)\ndim(\n,1\n3\n)\ndim(\n \n \n \n \n \n \nm\nz\nn\nx\ny\nn\nt\n ,\n \n)\n,\n(\n)\n,\n('\n \n \n \n \n \n \n \ny\nx\npol\nt\npol\n \n \n ,\n the region : \n \n \n \n \n \n \n \n \n \n3,2\n,\n0\n1\n)\n,\n(\n,\n0\n1\n)\n,\n(\n1\n)\n(\n~\n2\ns\ny\nx\nC\ny\nx\nC\nx\nC\nn\nn\ns\ns\nn\nn\nn\n \n \n \n \n \nthe parameterization : \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n3\n,...,\n,2,1\n,\n)\n)\n,\n(\n)(\n,\n(\n)\n,\n(\n2,1\n,\n1\n)\n,\n(\n)\n(\n1\n)1\n(\n]\n[\nq\ns\nq\ny\nx\nC\nx\nz\nC\nu\nz\nC\ns\ny\nz\nC\ns\nq\ns\nq\nm\ns\ns\ns\n \n \n \n \n \n \n(\n3\n2, \n \n are scalar constants)"},{"page":17,"text":"Theorem VII :\n))\n0\n0\n1\n1\n0\n0\n,\n0\n1\n0\n0\n1\n1\n,\n1\n1\n1\n)\n/\n1(\n,\n,\n1\n1\n1\n1\n1\n(\n(\nlim\n)\n,\n,\n,\n1\n1\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n*\n)\ndim(\n)\ndim(\n)\ndim(\n2\n0\n)\ndim(\n)\ndim(\n~\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nw\nw\nv\nv\nu\nu\nw\nw\nv\nv\nu\nu\nv\nv\nu\nw\nv\nv\nu\nu\nw\nv\nu\nv\nu\nt\nv\nv\nu\nh\nh\nv\nv\nu\nu\ngen\nt\nh\nv\nu\ngen\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nTheorem VIII :\n))\n,\n,\n,\n1\n1\n,\n1\n(\n(\nlim\n)\n,\n,\n,\n,\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n0\n \n \n \n \n \n \nt\nh\nw\nw\nz\ngen\nt\nh\nw\nz\ngen\nu\nu\nu\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nTheorem IX :\n)\n(\n))\n,\n,\n,\n,\n,\n(\n(\nlim\n)\n(\n)\n,\n,\n,\n0\n,\n(\n)\ndim(\n)\ndim(\n2\n0\n)\ndim(\n1\n)\ndim(\n~\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \ny\nt\nx\nh\nw\nz\ngen\ncoef\nt\nh\nw\nz\ngen\nh\nw\nx\nj\nj\nu\n \nwhere is a formal scalar variable, and for \n3,2,1\n,2,1\n \n \nq\ns\n:\n(i) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\ndim(\n)\ndim(\n)\ndim(\n3\n1\n0\n)1\n(\n)\n2\n(\n)\n(\n)1\n(\n)\n,\n(\n)\n,\n(\n)\n(\nz\nt\nw\ns\ns\nq\ns\nq\ns\nq\ns\nx\nt\nz\nC\nx\nw\nC\n \n \n \n \n \n \n \n \n(ii) \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n1\n(\n)\ndim(\n)\ndim(\n)\ndim(\n3\n)1\n(\n0\n)1\n(\n)\n2\n(\n)\n(\n)\n,\n(\n)\n,\n(\n)\n,\n(\n)1\n(\n)\n(\n \n \n \n \n \n \n \n \nq\nq\ns\nq\ns\ny\nx\nC\ny\nh\nz\nC\ny\nw\nC\nz\nh\nw\ns\nq\ns\ns\n \nProof:\nAccording to the definition, \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n))\n,\n,\n,\n,\n,\n(\n(\n)\ndim(\n \n \n \n \ny\nt\nx\nh\nw\nz\ngen\ncoef\nh\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nJ\nJ\nw\nI\nJ\nI\nJ\nI\ny\nK\nx\nh\nJ\nI\nK\nh\nK\nK\nw\nz\nesum\nw\nz\ny\nt\nx\nh\nesum\n)\n)\n0,0\n(\n,\n(\n)\n)\n0,0\n(\n)\n0,\n(\n)\n,\n(\n)\n0,\n(\n)\n,0\n(\n)\n,0\n(\n,\n(\n|\n|\n)\ndim(\n\\\n\\\n\\\n|\n|\n|\n|\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n|\n|,"},{"page":18,"text":"and in such a case \n)\n)\n0,\n(\n)\n,\n(,\n(\n)\ndim(\n \n \n \n \n \ny\ny\n is a right prolongation. Then, according to the\nprolongation-derivative principle and the equality (ii), the first multiplier of the expression \nunder the summation signs is \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n)\n0,\n(\n)\n0,0\n,\n(\n)\n0,0\n,0\n(\n)1,\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n|\n|\n3\n2\n1\n1\n|\n|\n)\ndim(\n)\n1\n(\n)\ndim(\n)\n(\nJ\nI\nK\nx\nh\nJ\nI\nK\nw\nz\nt\nx\nh\nesum\n \n \n \n \n \n \n)\n)\n0,\n(\n)\n0,0\n,\n(\n)\n0,0\n,0\n(\n)1,\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n)\n,\n(\n)\n,0\n(\n,\n(\n1\n|\n|\n3\n2\n1\n1\n|\n|\n)\ndim(\n)\ndim(\n)\ndim(\n1\n)\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nI\nK\nx\nh\nJ\nI\nK\nx\nj\nj\nw\nz\nt\nx\nh\nesum\n \n \n \n \n \n \n \nThen \n)\n)\n,0\n(\n)\n,\n(,\n(\n)\ndim(\n \n \n \n \n \nx\nx\n is a left prolongation, and we can apply the \nprolongation-derivative principle again, this time together with the equality (i), \nhence receiving\n \n \n)\ndim(\n1\n1\nx\nj\nj\n \n)\n)\n0,0\n,\n(\n)\n0,0\n(\n)\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n|\n|\n2\n2\n1\n|\n|\n)\ndim(\n)\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nI\nK\nh\nJ\nI\nK\no\nw\nz\nt\nh\nesum\n \n \n \n \nBecause the second multiplier doesn’t depend on , we can apply the operator \n...)\n(\nlim\n)\ndim(\n2\n0\nw\n \n \nto the first one only, hence receiving\n))\ndim(\n|\n(|\n)\n)\n0,0\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n))\ndim(\n|\n(|\n)\n)\n0,0\n,\n(\n)\n0,0\n(\n)\n0,0\n(\n)\n0,\n(\n)\n,0\n(\n,\n(\n)\ndim(\n1\n)\ndim(\n|\n|\n)\ndim(\n)\ndim(\n1\n)\ndim(\n|\n|\n)\ndim(\n)\n(\nw\nJ\nw\nz\nt\nh\nesum\nw\nJ\nw\nz\nt\nh\nesum\nx\nj\nj\nw\nI\nK\nh\nI\nK\nx\nj\nj\nw\nI\nK\nh\nI\nK\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nwhere, taking into account that the third sum\n \n \n)\ndim(\n|\n|,\nh\nK\nK\nruns over sets of cardinality \n)\ndim(h (thus “balancing” the patterns \n)\ndim(\n)\n,0\n(\nh\n \n and \nK\n \n)\n0,\n( \n), the last passage \nis due to the requirement (following from the E-sum’s definition) that the total"},{"page":19,"text":"balance is to be 0. So, after subjecting the product of the two multipliers to the \noverall summation \n \n \nI\nJ\nh\nK\nK\n)\ndim(\n|\n|,\n, we get the theorem’s claim. \nTheorem X (for any characteristic) :\n1)\n \n \n \n \n \n \n \nP\nj\ni\nj\ni\nz\nPart\nP\nz\nz\nz\nC\nper\n}\n,\n{\n2\n)})\ndim(\n,...,\n1\n({\n)\n(\n1\n))\n(\n~\n(\n2\n ,\nwhere for a set N\n)\n(\n2 N\nPart\nis the set of its perfect matchings. \n2) Let \n)\ndim(\n)\ndim(\ny\nx \n. Then\n)\n(\n)\n(\n)\ndim(\n1\n)\n,\n(\n)\n,\n(\n)\n(\n~\n))\n,\n(\n(\n))\n,\n,\n(\n~\n(\n)\n(\nz\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nz\ny\nx\nC\nper\n \n \n \n \n \nProof:\n1) first let’s prove that, if for a \nd\nd \n-matrix A we define \n \n \n \n \n \nd\nH\nd\ni\ni\ni\ndef\na\nA\nham\n \n \n1\n)\n(\n,\n)\n(\nwhere \nd\nH is the set of d -permutations with one cycle, \n0\n))\n(\n~\n(\n \nu\nC\nham\nwhen \n2\n)\ndim(\n \nu\n.\nSuppose \n2\n)\ndim(\n \nu\n. Let’s partition the set \nd\nH on disjoint subsets consisting of d -1 \nd -cycles which differ from each other only in the position of the element d . Each of those \nsubsets can be received via taking a ( d -1)-cycle \n)\n,...,\n(\n1\n1\n \n \ndh\nh\nh\n with elements from the set \n}\n1\n,...,\n1\n{\n \nd\n and alternate placing d between neighbors in h (altogether there are d -1 options).\nIn such a case the sum of the corresponding d -1 summands in the expression \n))\n(\n~\n(\nu\nC\nham\n will be\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n1\n1\n1\n1\n1\n1\n1\n1\n0\n)\n1\n1\n(\n)\n1\n(\n1\n)\n)(\n(\n)\n(\n)\n(\n)\n(\n)\n(\n)\n(\nd\ni\nd\nk\nd\nh\nd\nh\nd\ni\nh\nh\nh\nh\nd\nk\nh\nd\nd\nh\nh\nh\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nu\nk\nnext\nk\ni\nnext\ni\ni\nnext\ni\nk\nnext\nk\nk\nnext\nk\nwhere \n \n \n \n \n \n \n \n \n1\n,1\n1\n,1\n)\n(\nd\nt\nd\nt\nt\nt\nnext\n . \nTaking into account that for an \nn\nn \n-matrix A \n \n \n \n \n \n})\n,...,\n1\n({\n)\n,\n(\n)\n(\n)\n(\nn\nPart\nP\nP\nI\nI\nI\nA\nham\nA\nper\n \nand \n)\n(\n~\n))\n(\n~\n(\n)\n,\n(\nI\nI\nI\nz\nC\nz\nC\n \n, we get the first claim of the theorem;\n2) we apply the induction on \n)\ndim(z . \nFirst let’s prove the induction’s basis for \n1\n)\ndim(\n \nz\n. Let z be scalar. Then"},{"page":20,"text":")\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n1\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n(\nlim\n)))\n,\n(\n(\n1\n))\n,\n(\n(\n(\nlim\n))\n,\n,\n(\n~\n(\n2\n2\n0\n0\ny\nx\nC\ny\nx\nC\nz\ny\nz\nx\nC\nz\ny\nz\nx\nC\ny\nx\nC\nper\nz\ny\nz\nx\nC\nper\nz\ny\nx\nC\nper\n \n \n \n \n \n \n \n)\n))\n,\n(\ndet(\n))\n,\n(\ndet(\n1\n))\n(\n1\n))\n,\n(\n)\n,\n(\n(\n1\n))(\n,\n(\ndet(\n1\n)1(\n))\n,\n(\ndet(\n1\n(\nlim\n2\n2\n)\ndim(\n2\n2\n0\ny\nx\nC\ny\nx\nC\nO\ny\nz\nC\nx\nz\nC\ny\nx\nC\nO\ny\nx\nC\nx\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n(the second passage is due to the Borchardt formula).\nAnd now the induction’s step:\nlet \n1\n,\n)\ndim(\n \n \nm\nz\nm\nz\nis a scalar. Then\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n)\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\n1\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\nlim\n)\n))\n,\n,\n(\n~\n(\n1\n))\n,\n,\n(\n~\n(\n(\nlim\n))\n,\n,\n(\n~\n(\n)\ndim(\n)\ndim(\n1\n1\n1\n1\n0\n1\n1\n0\n1\nz\nz\nm\nm\nm\nm\nm\nm\nm\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nz\ny\nz\nC\nz\nx\nz\nC\nDiag\nz\nC\nper\nz\ny\nz\nx\nC\nper\nz\ny\nx\nC\nper\nz\nz\ny\nz\nx\nC\nper\nz\nz\ny\nx\nC\nper\n \n \n \n \n \n \n \n \n \n)\n(\n(\n)\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n,\n(\n(\n1\n))\n(\n))\n0\n1\n))\n,\n(\n)\n,\n(\n(\n(\n)\n(\n~\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n(\n1\n))\n,\n(\n)\n,\n(\n(\n1\n))(\n,\n(\n(\n1\nlim\n)\ndim(\n2\n)\ndim(\n1\n)\ndim(\n2\n)\ndim(\n1\n1\n0\nz\nz\nm\nz\nx\nm\nm\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\ny\nx\nC\nper\nO\ny\nz\nC\nx\nz\nC\nDiag\nz\nz\nC\nper\ny\nz\nC\nx\nz\nC\nDiag\nz\nC\nper\nO\ny\nz\nC\nx\nz\nC\ny\nx\nC\nper\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nThe last passage is due to the part (1) of this theorem.\nTheorem XI :\n))\n,\n,\n,\n1\n1\n0\n,\n1\n1\n1\n1\n1\n1\n(\n(\nlim\n)\n,\n,\n,\n(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n)\ndim(\n2\n0\n^\n \n \n \n \n \n \nt\nh\nu\nu\nu\ngen\nt\nh\nu\ngen\nu\nu\nu\nu\nu\nu\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nProof:"},{"page":21,"text":")\nX(2)\nTheorem\nuse\n(we\n))\n,\n,\n(\n~\n(\n))\n(\n~\n(\n)\n1\n(\n)\n,\n,\n,\n1\n1\n,\n(\n))\n,\n,\n,\n1\n1\n0\n,\n1\n1\n1\n1\n1\n1\n(\n(\nlim\n)\ndim(\n|\n|,\n\\\n|\n|\n)\ndim(\n~\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n)\ndim(\n~\n)\ndim(\n2\n0\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nI\nJ\nj\nJ\nh\nJ\nJ\nI\nJ\nI\nI\nu\nu\nu\nu\nu\nu\nu\nu\nt\nh\nC\nper\nu\nC\nper\nt\nh\nu\ngen\nt\nh\nu\nu\nu\ngen\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nJ\nj\nJ\nJ\nh\nJ\nJ\nu\nJ\nI\nJ\nj\nJ\nh\nJ\nJ\nJ\nu\nJ\nI\nI\nI\nI\nI\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nDiag\nper\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nDiag\nu\nC\nper\nu\nC\nper\n \n \n))\n,\n(\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n(\n)\nX(1)\nTheorem\nuse\nwe\n(\n))\n,\n(\n(\n))\n1\n))\n,\n(\n)\n,\n(\n((\n)\n(\n~\n(\n))\n(\n~\n(\n)\n1\n(\n)\ndim(\n|\n|,\n)\ndim(\n)\ndim(\n|\n|,\n)\ndim(\n\\\n\\\n\\\n|\n|\n \n \n)\n,\n,\n,\n(\n))\n,\n(\n(\n))\n1\n)\n,\n(\n1\n)\n,\n(\n(\n(\n^\n)\ndim(\n1\n)\ndim(\n)\ndim(\n)\ndim(\n|\n|,\n \n \nt\nh\nu\ngen\nt\nh\nC\nper\nt\nu\nC\nh\nu\nC\nu\ni\nJ\nj\nJ\nJ\nu\nJ\ni\nu\ni\nh\nJ\nJ\n \n \n \n \n \n \n \n \n \n \n \n \nTheorem XII :\nLet \n.\n0\n)\n(\n)\n,\n(\n),\ndim(\n)\ndim(\n)\ndim(\n2\n)\n1,\n3\n1\n(\nh\nDiag\nt\nh\nC\nh\nu\n \n \n \n \n \n Then\n)\n1\n,\n,\n,\n1\n(\n)\n1\n1(\n))\n,\n(\n)\n(\n)\n,\n(\n(\n3\n^\n)\ndim(\n3\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nt\nt\nh\nu\ngen\nh\nt\nC\nDiag\nt\nu\nC\nper\nm\nm\nh\n \nProof:\nIt follows from the generalized Binet-Minc identity for \n)\n(\nˆ\nA\nr\ne\np\nB\n, due to the fact \nthat \n0\n)\n1\n(\n)\n,\n(\n,3\n|\n|,\n)\n(\n)})\ndim(\n2\n,...,\n1\n{,\n()\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nt\nI\nDiag\nt\nt\nh\nC\nscal\nI\nI\n \n \n(by the construction\nand the condition \n)\ndim(\n2\n)\n1,\n3\n1\n(\n0\n)\n(\n)\n,\n(\nh\nDiag\nt\nh\nC\n \n \n \n \n), while \n)\ndim(\n)\ndim(\nh\nu \n.\nCorollary XII.1 :\nLet \n,0\n)\n)\n,\n(\n)\n,\n(\ndet(\n),\n2\n(\n)\ndim(\n,\n3\n)\ndim(\n)\ndim(\n)\n,...,\n1(\n)\n1,\n3\n1\n(\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nt\nu\nC\nt\nh\nC\nn\nn\nt\nn\nh\nu\nn\nm\nM is an \nn\nn -matrix. Then\n)\n1\n1\n)\n...\n0\n)\n,\n(\n))\n,\n(\n(\n.........\n)\n,\n(\n))\n,\n(\n(\n)\n,\n(\n(,\n,\n,\n1\n(\n)\n1\n1(\n)\n(\n)\n},\n,...,\n1\n({\n)\n1\n},\n,...,\n1\n({\n2\n1\n)\n1\n,...,\n(\n)\n,...,\n1\n(\n1\n)\n1\n,...,\n(\n1\n)\n,...,\n1(\n)1,\n3\n1\n(\n3\n^\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nn\nn\nn\nn\nn\nn\nT\nn\nn\nn\nT\nn\nn\nM\nM\nt\nu\nC\nh\nu\nC\nt\nu\nC\nh\nu\nC\nt\nh\nC\nt\nt\nh\nu\ngen\nM\nper\nm"},{"page":22,"text":"References:\n1) L.G. Valiant. The Complexity of Computing the Permanent.\nTheoret. Comp. Sci. 8, 189-201, 1979.\n2) Okada Soichi. Generalizations of Cauchy’s Determinant Identity \nand Schur’s Pfaffian Identity.\nhttp://citeseer.ist.psu.edu/cache/papers/cs2/125/http:zSzzSzwww.math.kobe-\nu.ac.jpzSzpublicationszSzrlm18zSz9.pdf/generalizations-of-cauchy-s.pdf\n3) Richard A. Brualdi, Herbert J. Ryser, Combinatorial Matrix Theory, \nCambridge University Press, Cambridge, 1991. Comm. Computer Algebra \n39 (2005), pp. 61–64.\n 4) Henryk Minc, Permanents, Addison-Wesley, Reading, 1978.\n 5) http://en.wikipedia.org/wiki/Pfaffian\n 6) http://en.wikipedia.org/wiki/Cauchy_determinant\n 7) Mark Jerrum. The computational complexity of counting. \nThe International Congress of Mathematicians, Zurich, August 1994.\nhttp://www.lfcs.inf.ed.ac.uk/reports/94/ECS-LFCS-94-296/index.html"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"P and NP what is equivalent to RP=NP( Ref. [1]).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"}],"equation_refs":[]},"symbol_table":[],"references":{"in_text_markers":[],"bibliography":[]},"claims":[],"flow_graph":{"nodes":[],"edges":[]},"quality":{"text_char_count":36706,"parse_confidence":0.5,"equation_parse_rate_proxy":0.05,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}