{"paper_meta":{"paper_id":"arxiv:0710.2732","title":"0710.2732","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":"arXiv:0710.2732v1 [cs.CC] 15 Oct 2007\nProbabilistic communication complexity over the reals\nDima Grigoriev\nCNRS, IRMAR, Universit ́e de Rennes\nBeaulieu, 35042, Rennes, France\ndmitry.grigoryev@univ-rennes1.fr\nhttp://perso.univ-rennes1.fr/dmitry.grigoryev\nAbstract\nDeterministic and probabilistic communication protocols are introduced in which par-\nties can exchange the values of polynomials (rather than bits in the usual setting). It\nis established a sharp lower bound 2n on the communication complexity of recognizing\nthe 2n-dimensional orthant, on the other hand the probabilistic communication complex-\nity of its recognizing does not exceed 4. A polyhedron and a union of hyperplanes are\nconstructed in R2n for which a lower bound n/2 on the probabilistic communication com-\nplexity of recognizing each is proved. As a consequence this bound holds also for the\nEMPTINESS and the KNAPSACK problems.\nIntroduction\nCommunication complexity (see [15], a survey one can find in [12], [13]) in the usual (bit)\nsetting counts the number of bit exchanges between two (or more) parties who altogether\ncompute a certain function (one of the goals of the communication complexity was to pro-\nvide a framework to analyze distributed computations and to obtain lower bounds on other\ncomplexity ressources). In [2] one can find the relations of the communication complexity\nwith the question of representing a function as a composition of functions of a special form\n(this question stems from the Hilbert’s 13th problem). In [5] the communication complexity\nof quantum computations was studied.\nIn the present paper we introduce the model of communication protocols over real (or com-\nplex) numbers when the parties exchange the values of polynomials. The variables of polyno-\nmials are supposed to be partitioned in two groups: X = {X1, . . . , Xn1}, Y = {Y1, . . . , Yn2},\nthe first party is able to calculate polynomials in X, the second party in Y . It is worthwhile\nto mention that in [11] a different (less restrictive) concept of a communication protocol was\nintroduced in which the parties can exchange arbitrary real numbers (rather than just val-\nues of a given family of polynomials as in the present paper). After the present paper had\nbeen submitted the paper [3] has appeared in which a similar algebraic communication pro-\ntocol was introduced and several lower bounds on the algebraic communication complexity\nfor computing rational functions and recognizing algebraic varieties were established. Unlike\n[3] we obtain lower bounds on probabilistic communication complexity and in addition, for\nrecognizing real semi-algebraic sets.\nWe note that parallel to the numerous customary (boolean or discrete) complexity classes\none develops also their continuous (algebraic or semi-algebraic) counterparts (see e. g. [4],\n[6]). This paper presents an attempt to introduce and study the probabilistic continuous\ncommunication complexity.\n1\n\nFor illustration of the results obtained in the present paper we consider the KNAP-\nSACK problem: whether for given sets {x1, . . . , xn} and {y1, . . . , yn} there exist subsets\nI1, I2 ⊆{1, . . . , n} such that P\ni1∈I1 xi1 + P\ni2∈I2 yi2 = 0? There is an evident deterministic\ncommunication protocol for the KNAPSACK problem with the communication complexity\n2n when two parties just yield {x1, . . . , xn} and {y1, . . . , yn}, respectively. In Section 4 we\nshow a lower bound n/4 in the complex case and n/2 in the real case on the probabilistic\ncommunication complexity for the KNAPSACK problem.\nIn Section 1 we define the communication complexity of computing a function (polynomial\nfor simplicity) and show a lower bound on it being the rank of the matrix of its second\nderivatives, earlier this matrix in the frames of communication complexity was employed in\n[1]. This slightly resembles the lower bound on the bit communication complexity being the\nlogarithm of the rank of the communication matrix [15].\nIn Section 2 we describe the (deterministic) communication protocols (respectively, proba-\nbilistic communication protocols) and relying on this we define the (deterministic) communi-\ncation complexity of recognizing a set (respectively, probabilistic communication complexity).\nAs an application of the matrix of the second derivatives we establish a lower bound n −3\non a probabilistic communication complexity of recognizing a constructible set in C2n whose\nZariski closure contains the hypersurface {f = X1Y1 +· · ·+XnYn = 0}. As a real counterpart\nwe establish the same bound n −3 for a semialgebraic set in R2n whose euclidean closure has\n(full) (2n −1)-dimensional intersection with the hypersurface {f = 0}.\nIn Section 3 we demonstrate a possible exponential gap between the deterministic and\nprobabilistic communication complexities. Namely, we prove a (sharp) lower bound n1 + n2\non the deterministic communication complexity of recognizing the orthant\n{(x1, . . . , xn1, y1, . . . , yn2) ∈Rn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nOn the other hand, we show that the probabilistic communication complexity of recognizing\nthe orthant does not exceed 4.\nIn Section 2 the lower bound was established for a set which involves a polynomial f\nwith a big communication complexity of its computation.\nIn Section 4 we construct sets\ndefined by linear contraints which nevertheless have big probabilistic communication com-\nplexity (clearly, any linear function has the communication complexity of its computation\nat most 2). Namely, we consider the polyhedron {Xi + Yi > 0, 1 ≤i ≤n} ⊂R2n and the\narrangement ∪1≤i,j≤n{Xi + Yj = 0} ⊂R2n and for each of both prove a lower bound n/2\non the probabilistic communication complexity of its recognizing. For the complex arrange-\nment ∪1≤i,j≤n{Xi + Yj = 0} ⊂C2n we establish a lower bound n/4. As applications the\nobtained lower bounds imply the same bounds for the EMPTINESS problem, i. e whether\n{x1, . . . , xn} ∩{y1, . . . , yn} = ∅, and for the KNAPSACK problem.\n1\nLower bound on the communication complexity of comput-\ning a function\nFirst we describe computational models for the communication complexity over com-\nplex\nor\nreal\nnumbers.\nLet\ntwo\nfamilies\nof\nvariables\nX\n=\n{X1, . . . , Xn1}\nand\nY\n=\n{Y1, . . . , Yn2}\nbe\ngiven.\nAs\nusually\nin\ncommunication\ncomplexity\nstudies,\nthere are two parties.\nWe assume that one party is able to calculate polynomials\na1(X), . . . , ar1(X) in X and the second party is able to calculate polynomials b1(Y ), . . . , br2(Y )\n2\n\nin Y .\nThen the result is obtained by means of calculating suitable polynomials\nP1(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )), . . . , PN(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )). The\ngoal is to minimize r1 + r2 viewed as a measure of communication complexity.\nWe study the communication complexity of two problems:\ncomputing a polynomial\ng(X, Y ) and recognizing a subset S in (n1 + n2)-dimensional complex or real space.\nDefinition 1.1 A polynomial g(X, Y ) has a communication complexity c(g) less or equal\nto r1 + r2 if g\n=\nP(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )) for appropriate polynomials\nP, a1, . . . , ar1, b1, . . . , br2.\nObviously, the communication complexity of g does not exceed n1 + n2.\nBy H(g) denote n1 ×n2 matrix of the second derivatives (\n∂2g\n∂Xi∂Yj ), by H(P) denote r1 ×r2\nmatrix (\n∂2P\n∂ai1∂bj1 ), by the Jacobian J(a1, . . . , ar1) denote n1 × r1 matrix of the first derivatives\n(\n∂ai1\n∂Xi ), similar J(b1, . . . , br2) = (\n∂bj1\n∂Yj ). Then we have\nH(g) = J(a1, . . . , ar1)H(P)(J(b1, . . . , br2))T .\nLemma 1.2 (cf. [1]) In the notations of Definition 1.1 we have\nc(g) ≥min{r1, r2} ≥rk(H(g)).\nCorollary 1.3 c(f = X1Y1 + · · · + XnYn) ≥n\nTo deal in the sequel with communication protocols we need the following statement\ngeneralizing the latter corollary.\nLemma 1.4 Let a polynomial g be a multiple of f. Then rk(H(g)) ≥n −3.\nProof. We write g = f mh where f does not divide h (evidently, f is absolutely irredicible\nwhen n ≥2, we assume here that n1 = n2 = n). We have\nH(g) = mf m−1h\n \n∂2f\n∂Xi∂Yj\n \n+ f m\n \n∂2h\n∂Xi∂Yj\n \n+\nm(m −1)f m−2h\n ∂f\n∂Xi\n ∂f\n∂Yj\n \n+ mf m−1\n ∂f\n∂Xi\n ∂h\n∂Yj\n \n+ mf m−1\n ∂h\n∂Xi\n ∂f\n∂Yj\n \n.\nEach of the latter three matrices has rank at most 1, so it suffices to verify that the sum of\nthe former two matrices divided by f m−1 is non-singular, it equals\nM = mh\n \n∂2f\n∂Xi∂Yj\n \n+ f\n \n∂2h\n∂Xi∂Yj\n \nWe have det(M) = (mh)n + ff1 for a certain polynomial f1, hence det(M) ̸= 0.\nIt would be interesting to clarify, whether one can majorate c(g) via an appropriate\nfunction in rk(H(g))?\n3\n\n2\nProbabilistic communication protocols\nNow we define a communication protocol for recognizing a set S. We consider two cases:\nS ⊂Cn1+n2 is a constructible set or S ⊂Rn1+n2 is a semialgebraic set.\nA protocol is\na rooted tree, and to its root an input (x, y) = (x1, . . . , xn1, y1, . . . , yn2) is attached.\nTo\nevery vertex v of the tree (including the root, but excluding the leaves) either a certain\npolynomial av(X) or a polynomial bv(Y ) is attached (so, it is calculated either by the first\nparty or by the second party, respectively). To every vertex v (of a depth r) leads a unique\npath from the root, denote by q1(X, Y ), . . . , qr(X, Y ) the polynomials attached to the ver-\ntices v1, . . . , vr = v along this path, thus for every 1 ≤k ≤r either qk(X, Y ) = avk(X)\nor qk(X, Y ) = bvk(Y ), respectively.\nIn addition, to the vertex v a family of testing poly-\nnomials Pv,1(Q1, . . . , Qr), . . . , Pv,Nv(Q1, . . . , Qr) is assigned.\nSimilar to the usual decision\ntrees (see\ne.g.\n[14], [9], [10]) the protocol ramifies at v according to the set of the\nsigns sgn(Pv,1(q1(x, y), . . . , qr(x, y))), . . . , sgn(Pv,Nv(q1(x, y), . . . , qr(x, y))).\nSimilar to deci-\nsion trees in the complex case the sign can attain two values: =, ̸=, in the real case three\nvalues: =, <, >. To every leaf a label either “accept” or “reject” is assigned which provides an\noutput of the protocol. To the protocol naturally corresponds a decision tree (without restric-\ntions on the degrees of testing polynomials). To any input (x, y) corresponds a unique leaf\nof the protocol and a path leading to this leaf, according to the signs of testing polynomials:\nthe output assigned to the leaf is “accept” if and only if (x, y) ∈S.\nThe communication complexity of the recognizing protocol is defined as its depth. We note\nthat the communication complexity counts just the number of the polynomials avi(X) or\nbvi(Y ), respectively, calculated (separately) by each of both parties in several rounds along a\npath of the protocol and ignores the (jointly) calculated polynomials Pv,1, . . . , Pv,Nv.\nNow we introduce probabilistic communication protocols.\nOne can define it similar to\nprobabilistic decision trees (cf. [14], [9], [8], [10]) as a finite family C = {Ci}i of communication\nprotocols Ci, chosen with a certain probability pi ≥0, where P\ni pi = 1. As for decision trees\nwe require that a probabilistic communication protocol for any input returns a correct output\nwith the probability greater than 2/3 (we suppose that a certain continuous probabilistic\nmeasure is fixed in the ambient space,\ne.g.\none can take the Gaussian measure).\nThe\nmaximal depth of communication protocols which constitute a probabilistic communication\nprotocol is called the probabilistic communication complexity.\nFirst we consider probabilistic communication protocols over complex numbers.\nProposition 2.1 The probabilistic communication complexity of an (2n −1)-dimensional\nconstructible set W ⊂C2n such that its Zariski closure W contains the hypersurface U =\n{f = X1Y1 + · · · + XnYn = 0} is greater or equal to n −3.\nProof. Let a probabilistic communication protocol C recognize W. Among communica-\ntion protocols which constitute C there exists C0 such that it gives the correct outputs for at\nleast of 1/3 of the points from U and for at least of 1/3 of the points outside of U (in fact,\nfor the arguments below, instead of 1/3 any positive constant would suffice).\nDistinguish in the decision tree corresponding to C0 a (unique) path along which all the\nsigns in the ramifications are ̸=. Denote by {Pj(q1(x, y), . . . , qr(x, y))}1≤j≤N the collection\nof all the testing polynomials along this path, clearly r does not exceed the communication\ncomplexity of C0. Denote P = Q\n1≤j≤N Pj(q1, . . . , qr). Then the inputs from the Zariski-open\nset V = {(x, y) : P(x, y) ̸= 0} ⊂C2n follow this path in C0.\n4\n\nDue to the choice of C0 we conclude that f divides P. Indeed, C0 rejects all the points\nfrom a suitable (constructive) subset of C2n of the dimension 2n because C0 rejects a subset\nof a positive (namely, at least 1/3) measure, whence if f did not divide P then C0 would\nreject all the points of U except for its certain (constructive) subset of the dimension at most\n2n −2, but on the other hand, C0 should accept a subset of a positive measure (at least 1/3)\nfrom U. Therefore, Lemma 1.4 and Lemma 1.2 imply that r ≥c(P) ≥rk(H(P)) ≥n −3.\nFor a semialgebraic set S ⊂Rn1+n2 denote by ∂(S) ⊂Rn1+n2 its boundary, being a\nsemialgebraic set as well. The following proposition is a real counterpart of Proposition 2.1.\nCorollary 2.2 The probabilistic communication complexity of a semialgebraic set S such that\ndim(∂(S) ∩U) = 2n −1 is greater or equal to n −3.\nProof. For any communication protocol Ci from C consider the product P (Ci) =\nQ\n1≤j≤N Pj of all the testing polynomials from Ci (cf. the proof of Proposition 2.1 where\na similar product of the polynomials along a particular path was taken).\nFor any point\nu ∈∂(S) ∩U there exists C0 such that P (C0)(u) = 0, otherwise all the points from an appro-\npriate ball centered at u would get the same output for all communication protocols Ci from\nC which would contradict the definition of the boundary. Hence there exists C0 for which f\ndivides P (C0). Therefore, we complete the proof as at the end of Proposition 2.1.\n3\nCommunication complexity of recognizing the orthant\nNow we proceed to estimating the communication complexity of the orthant T\n=\n{(x1, . . . , xn1, y1, . . . , yn2) ∈Rn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nFor this\ngoal we use infinitesimals ǫ1 > · · · > ǫn1+n2 > 0 (see e.g.[7], [9], [8], [10]). Namely, denote by\nRi =\n^\nR(ǫ1, . . . , ǫi) by recursion on i the real closure of the field R(ǫ1, . . . , ǫi), for the base of\nrecursion we put R0 = R. Then ǫi+1 is transcendental over Ri and for any positive element\n0 < d ∈Ri we have 0 < ǫi+1 < d.\nFor a polynomial g ∈R[X1, . . . , Xn1, Y1, . . . , Yn2] denote by lt(g) its least term with respect\nto the following (lexicographical) ordering: take the terms with a minimal degree in Yn2,\namong them with a minimal degree in Yn2−1 and so on. If lt(g) = g0Xi1\n1 · · · X\nin1\nn1 Y j1\n1 · · · Y\njn2\nn2\nfor a certain g0 ∈R, we call (i1, . . . , in1, j1, . . . , jn2) the exponent vector of lt(g).\nTake\ne1, . . . , en1+n2 ∈{−1, 1}, then we have (cf. [9], [8], [10])\nsgn(g(e1ǫ1, . . . , en1+n2ǫn1+n2)) = sgn(lt(g)(e1ǫ1, . . . , en1+n2ǫn1+n2))\n(1)\nLemma 3.1 Let g1, . . . , gs ∈R[X1, . . . , Xn1, Y1, . . . , Yn2] and P1, . . . , PN ∈R[G1, . . . , Gs].\nThen among the exponent vectors of the least terms of P1(g1, . . . , gs), . . . , PN(g1, . . . , gs) there\nare at most s linearly independent.\nProof. We claim that if exponent vectors of any family of polynomials h1, . . . , ht ∈\nR[X1, . . . , Xn1, Y1, . . . , Yn2] are linearly independent then h1, . . . , ht are algebraically indepen-\ndent over R. Indeed, denote the exponent vectors of lt(h1), . . . , lt(ht) by l1, . . . , lt, respectively,\nand denote by L the t × (n1 + n2) matrix with the rows l1, . . . , lt, then for any polynomial\nP = P\nK pKGK ∈R[G1, . . . , Gt] the exponent of the least term of P(h1, . . . , ht) coincides with\nthe least vector among the pairwise distinct vectors KL for all K ∈Zt such that pK ̸= 0.\nThe proved claim entails the lemma immediately.\n5\n\nTheorem 3.2 The communication complexity of recognizing the orthant T (as well as its\nclosure T in the euclidean topology) is greater or equal to n1 + n2.\nProof. Let a communication protocol C0 recognize T (the arguing for T is similar). Using\nthe Tarski’s transfer principle (see e. g. [7], [9], [8], [10]) one can extend the inputs of C0\nover the field Rn1+n2, then C0 recognizes the set T (Rn1+n2) = {(x1, . . . , xn1, y1, . . . , yn2) ∈\nRn1+n2\nn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nTake in C0 the path which follows\nthe input (ǫ1, . . . , ǫn1+n2) ∈T (Rn1+n2).\nLet r be the length of this path and denote by\nq1(X, Y ), . . . , qr(X, Y ) the polynomials attached to the vertices along this path (we use the\nnotations introduced in Section 2 and recall that every qi depends either on X or on Y ,\nalthough the latter is not used in the proof of the Theorem, cf. Remark 3.3 below). Let\nP1(q1, . . . , qr), . . . , PN(q1, . . . , qr) be all the testing polynomials along this path.\nLemma\n3.1\nimplies\nthat\namong\nthe\nexponent\nvectors\nof\nlt(P1(q1, . . . , qr)), . . . , lt(PN(q1, . . . , qr))\nthere\nare\nat\nmost\nr\nlinearly\nindependent\nK1, . . . , Kr0, r0 ≤r.\nSuppose that the theorem is wrong and r < n1 + n2.\nPick a\nboolean vector 0 ̸= (m1, . . . , mn1+n2) ∈(Z/2Z)n1+n2 orthogonal to all Ki(mod 2), 1 ≤i ≤r0.\nThen\nsgn(Pj(q1, . . . , qr)(ǫ1, . . . , ǫn1+n2)) = sgn(Pj(q1, . . . , qr)((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2))\nfor 1 ≤j ≤N (cf. the proof of lemma 1 [9]). This means that the output of C0 is the same for\nthe inputs (ǫ1, . . . , ǫn1+n2) and ((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2). The obtained contradiction\nwith the supposition completes the proof of the theorem.\nRemark 3.3 The bound in Theorem 3.2 still holds if instead of communication protocols\none considers more general decision trees omitting the condition that each of the polynomials\nq1(X, Y ), . . . , qr(X, Y ) depends either on X or on Y . This strengthens slightly lemma 1 [9]\nsince here we consider decision trees without a priori bound on fan-out of branching, unlike\n[9] where the fan-out did not exceed 3.\nRemark 3.4 Clearly, the communication complexity in the theorem equals n1 + n2.\nRemark 3.5 The probabilistic communication complexity of recognizing the closure T\ndoes not exceed logO(1)(n1 + n2).\nIndeed, the first party tests whether for an input\n(x1, . . . , xn1, y1, . . . , yn2) the inequalities x1 ≥0, . . . , xn1 ≥0 hold by means of a probabilis-\ntic decision tree of the depth logO(1) n1 due to Theorem 1 [9]. The second party tests the\ninequalities y1 ≥0, . . . , yn2 ≥0 by the same token.\nThe latter remark demonstrates an exponential gap between the probabilistic and de-\nterministic communication complexities for recognizing the closure T. The next proposition\nprovides even a bigger gap for T.\nProposition 3.6 The probabilistic communication complexity of recognizing T is at most 4.\nProof. For an input (x1, . . . , xn1, y1, . . . , yn2) consider the partition of the indices\n{1, . . . , n1 + n2} = I0 ∪I+ ∪I−into the subsets for which the corresponding coordinates\nof the input are zero, positive or negative, respectively. If I0 ∪I−̸= ∅then for a randomly\n6\n\nchosen subset I ⊆{1, . . . , n1+n2} the probability of the event that I ∩I0 = ∅and that |I ∩I−|\nis even is less or equal to 1/2. The latter statement is obvious when I0 ̸= ∅, and when I0 = ∅\nthis probability equals to 1/2.\nTherefore, when (x1, . . . , xn1, y1, . . . , yn2) /∈T and if one chooses randomly a product\nQ\ni1∈I1 xi1\nQ\ni2∈I2 yi2 then this product is positive with the probability less or equal to 1/2.\nThus, the first party chooses randomly independently two subsets I(1), I(2) ⊆{1, . . . , n1} and\ncalculates the products Q\ni∈I(1) xi and Q\ni∈I(2) xi (in a similar way the second party). If all 4\ncalculated products are positive then the output is “accept”, otherwise “reject”.\n4\nLower bound on probabilistic communication complexity\nCorollary 2.2 together with Lemma 1.4 show that if the (n1 +n2−1)-dimensional boundary of\na semialgebraic set contains a “facet” with a great communication complexity of computing\nthe polynomial which determines this facet, then the probabilistic communication complexity\nof recognizing this set is great as well. Now we construct a set (being a polyhedron) with a\ngreat probabilistic communication complexity (note that any facet of the polyhedron being\ndetermined by a linear function, has a communication complexity at most 2).\nConsider the polyhedron S = {(x1, . . . , xn, y1, . . . , yn) ∈R2n : xi + yi > 0, 1 ≤i ≤n}\nand an arrangement R either real (i. e. ⊂R2n) or complex (i. e. ⊂C2n) being a union of\nhyperplanes among which there appear n hyperplanes {Xi + Yi = 0}, 1 ≤i ≤n.\nTheorem 4.1 The probabilistic communication complexity of recognizing over the reals the\nset S or the set R is greater than n/2.\nProof.\nDenote\nZi\n=\nXi + Yi,\n1\n≤\ni\n≤\nn.\nWe\nconsider the new\nco-\nordinates\n(X1, . . . , Xn, Z1, . . . , Zn)\nin\nR2n\nand\nthe\npoint\nu\n=\n(ǫ1, . . . , ǫ2n).\nLet\na\nprobabilistic\ncommunication\nprotocol\nC\nrecognize\nS\n(respectively,\nR).\nIntro-\nduce\nn\npoints\nui\n=\n(ǫ1, . . . , ǫn+i−1, −ǫn+i, ǫn+i+1, . . . , ǫ2n)\n(respectively,\nu(0)\ni\n=\n(ǫ1, . . . , ǫn+i−1, 0, ǫn+i+1, . . . , ǫ2n)), 1 ≤i ≤n.\nClearly, u ∈S, ui\n/∈S (respectively,\nu /∈R, u(0)\ni\n∈R).\nThere exists a communication protocol C0 from the family constituting C which gives\ncorrect outputs for the input u and for at least of n/2 inputs among ui (respectively, u(0)\ni ).\nWithout loss of generality one can assume that the outputs are correct for all ui, 1 ≤i ≤⌈n/2⌉\n(respectively, for u(0)\ni ).\nTake the path in C0 which follows the input u and consider the testing polyno-\nmials P1(q1, . . . , qr), . . . , PN(q1, . . . , qr) along this path (cf.\nSection 2).\nDenote P\n=\nQ\n1≤j≤N Pj(q1, . . . , qr). We claim that the least term lt(P) = Q\n1≤j≤N lt(Pj(q1, . . . , qr)) di-\nvides on each Zi, 1 ≤i ≤⌈n/2⌉(recall that the least term is defined with respect to the\ncoordinates (X1, . . . , Xn, Z1, . . . , Zn)). Otherwise, if lt(P) does not divide on Zi then we have\nsgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(ui)), 1 ≤j ≤N\n(respectively,\nsgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(u(0)\ni ))).\nHence C0 gives the same output for both inputs u and ui (respectively, u(0)\ni ). The obtained\ncontradiction proves the claim.\n7\n\nThus, the theorem would follow from the next lemma taking into account Lemma 1.2.\nLemma 4.2 If for a certain k > 1 the product Z1 · · · Zk divides lt(P) then for the rank of\nn × n matrix we have\nrk\n \n∂2P\n∂Xi∂Yj\n \n≥k\nProof. Let lt(P) = p0Xm1\n1\n· · · Xmn\nn\nZl1\n1 · · · Zln\nn where p0 ∈R. Then the highest term (cf.\n(1)) of a non-diagonal entry\n∂2P\n∂Xi∂Yj (u) when i ̸= j, 1 ≤i, j ≤k equals\nliljǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫn+iǫn+j\nThe highest term of a diagonal entry\n∂2P\n∂Xi∂Yi (u) either equals\nli(li −1)ǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫ2\nn+i\nwhen li > 1 or is less than\nǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫ2\nn+i\n.\nDenote by M k × k matrix with the diagonal (i, i)-entries li(li −1) and the non-diagonal\n(i, j)-entries lilj, 1 ≤i, j ≤k. Then det(M) = (−1)k+1l1 · · · lk(l1 + · · · + lk −1) ̸= 0 when\nk > 1. Therefore, the coefficient of the k × k minor\ndet\n \n∂2P\n∂Xi∂Yj\n \n(u)\nwhere 1 ≤i, j ≤k at its highest term\n(ǫm1\n1\n· · · ǫmn\nn )kǫkl1−2\nn+1 · · · ǫkln−2\n2n\nequals to det(M) and thereby, it does not vanish, which proves the lemma.\nRemark 4.3 The same bound as in the theorem holds as well for the (euclidean) closure S.\nCorollary 4.4 The probabilistic communication complexity over complex numbers of R is\ngreater than n/4.\nProof. Having a probabilistic communication protocol C over C which recognizes R,\none can convert it into a probabilistic communication protocol C(R) over reals which rec-\nognizes R at the cost of increasing the complexity at most twice.\nFor this purpose the\nfirst party replaces every polynomial a(X) in C which the first party calculates by a pair\nof polynomials Re(a), Im(a) ∈R[X] in C(R) where a = Re(a) + √−1Im(a).\nThe same\nfor the second party.\nThen for each testing polynomial Pj(q1, . . . , qr) its real and imagi-\nnary parts Re(Pj(q1, . . . , qr)), Im(Pj(q1, . . . , qr)) can be expressed as polynomials over R in\nRe(ql), Im(ql), 1 ≤l ≤r. Any ramification condition Pj(q1, . . . , qr) = 0 in C we replace in\n8\n\nC(R) by Re(Pj(q1, . . . , qr)) = Im(Pj(q1, . . . , qr)) = 0. To complete the proof of the corollary\nwe apply Theorem 4.1 to C(R).\nAs particular cases consider the problem EMPTINESS: whether the intersection of two\nfinite sets {x1, . . . , xn} ∩{y1, . . . , yn} = ∅is empty?\nIt corresponds to the arrangement\n∪i,j{xi = yj} (in C2n or R2n).\nAnother example is the KNAPSACK problem: whether\nthere exist subsets I1, I2 ⊆{1, . . . , n} such that P\ni1∈I1 xi1 + P\ni2∈I2 yi2 = 0? It can be also\nrepresented as an arrangement (cf. [10]).\nCorollary 4.5 The probabilistic communication complexity of both EMPTINESS and\nKNAPSACK problems is greater than n/4 over C and greater than n/2 over R.\nAcknowledgements.\nThe author is grateful to the Max-Planck Institut fuer Math-\nematik, Bonn where the paper was written, to Farid Ablayev and to Harry Buhrman for\ninteresting discussions and to anonymous referees for very detailed comments, which helped\nto improve the presentation of the paper.\nReferences\n[1] H. Abelson, Lower bounds on information transfer in distributed computations, J. Assoc.\nComput. Mach., 27 (1980), 384–392.\n[2] F. Ablayev, S. Ablayeva, A discrete approximation and communication complexity ap-\nproach to the superposition problem, in Proc. Intern. Symp. Fundamentals of Computa-\ntion Theory, Lect. Notes Comput. Sci., 2138, (2001), Springer, 47–58.\n[3] M. Bl ̈aser, E. Vicari, Algebraic communication complexity, Preprint (2007).\n[4] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and real computations, Springer\n(1998).\n[5] H. Buhrman, R. de Wolf, Communication complexity lower bounds by polynomials, Proc.\nIEEE Conf. Computational Complexity (2001), 120–130.\n[6] P. B ̈urgisser, Completeness and reduction in algebraic complexity theory, Springer (2000).\n[7] D. Grigoriev, N. Vorobjov, Solving systems of polynomial inequalities in subexponential\ntime, J. Symb. Comput., 5 (1988), 37–64.\n[8] D. Grigoriev, M. Karpinski, F. Meyer auf der Heide, R. Smolensky, A lower bound for\nrandomized algebraic decision trees, Computational Complexity, 6 (1996/1997), 357–375.\n[9] D. Grigoriev, M. Karpinski, R. Smolensky, Randomization and the computational power\nof analytic and algebraic decision trees, Computational Complexity, 6 (1996/1997), 376–\n388.\n[10] D. Grigoriev, Randomized complexity lower bounds for arrangements and polyhedra, Dis-\ncrete Computational Geometry, 21 (1999), 329–344.\n[11] J. Krajiˆcek. Interpolation by a game, Math. Logic Quat., 44 (1998), 450–458.\n9\n\n[12] E. Kushilevitz, N. Nisan, Communication complexity, Cambridge (1997).\n[13] L. Lovasz, Communication complexity: a survey, in “Paths, flows and VLSI layout”,\nKorte, Lovasz, Proemel, Schrijver Eds. (1990), Springer, 235–266.\n[14] F. Meyer auf der Heide, Simulating probabilistic by deterministic algebraic computation\ntrees, Theor. Comp. Sci., 41 (1984), 325–330.\n[15] A. Yao, Some complexity questions related to distributive computing, in Proc. ACM Symp.\nTheory on Computing (1979), 209–213.\n10","paragraphs":[{"paragraph_id":"p1","order":1,"text":"arXiv:0710.2732v1 [cs.CC] 15 Oct 2007\nProbabilistic communication complexity over the reals\nDima Grigoriev\nCNRS, IRMAR, Universit ́e de Rennes\nBeaulieu, 35042, Rennes, France\ndmitry.grigoryev@univ-rennes1.fr\nhttp://perso.univ-rennes1.fr/dmitry.grigoryev\nAbstract\nDeterministic and probabilistic communication protocols are introduced in which par-\nties can exchange the values of polynomials (rather than bits in the usual setting). It\nis established a sharp lower bound 2n on the communication complexity of recognizing\nthe 2n-dimensional orthant, on the other hand the probabilistic communication complex-\nity of its recognizing does not exceed 4. A polyhedron and a union of hyperplanes are\nconstructed in R2n for which a lower bound n/2 on the probabilistic communication com-\nplexity of recognizing each is proved. As a consequence this bound holds also for the\nEMPTINESS and the KNAPSACK problems.\nIntroduction\nCommunication complexity (see [15], a survey one can find in [12], [13]) in the usual (bit)\nsetting counts the number of bit exchanges between two (or more) parties who altogether\ncompute a certain function (one of the goals of the communication complexity was to pro-\nvide a framework to analyze distributed computations and to obtain lower bounds on other\ncomplexity ressources). In [2] one can find the relations of the communication complexity\nwith the question of representing a function as a composition of functions of a special form\n(this question stems from the Hilbert’s 13th problem). In [5] the communication complexity\nof quantum computations was studied.\nIn the present paper we introduce the model of communication protocols over real (or com-\nplex) numbers when the parties exchange the values of polynomials. The variables of polyno-\nmials are supposed to be partitioned in two groups: X = {X1, . . . , Xn1}, Y = {Y1, . . . , Yn2},\nthe first party is able to calculate polynomials in X, the second party in Y . It is worthwhile\nto mention that in [11] a different (less restrictive) concept of a communication protocol was\nintroduced in which the parties can exchange arbitrary real numbers (rather than just val-\nues of a given family of polynomials as in the present paper). After the present paper had\nbeen submitted the paper [3] has appeared in which a similar algebraic communication pro-\ntocol was introduced and several lower bounds on the algebraic communication complexity\nfor computing rational functions and recognizing algebraic varieties were established. Unlike\n[3] we obtain lower bounds on probabilistic communication complexity and in addition, for\nrecognizing real semi-algebraic sets.\nWe note that parallel to the numerous customary (boolean or discrete) complexity classes\none develops also their continuous (algebraic or semi-algebraic) counterparts (see e. g. [4],\n[6]). This paper presents an attempt to introduce and study the probabilistic continuous\ncommunication complexity.\n1"},{"paragraph_id":"p2","order":2,"text":"For illustration of the results obtained in the present paper we consider the KNAP-\nSACK problem: whether for given sets {x1, . . . , xn} and {y1, . . . , yn} there exist subsets\nI1, I2 ⊆{1, . . . , n} such that P\ni1∈I1 xi1 + P\ni2∈I2 yi2 = 0? There is an evident deterministic\ncommunication protocol for the KNAPSACK problem with the communication complexity\n2n when two parties just yield {x1, . . . , xn} and {y1, . . . , yn}, respectively. In Section 4 we\nshow a lower bound n/4 in the complex case and n/2 in the real case on the probabilistic\ncommunication complexity for the KNAPSACK problem.\nIn Section 1 we define the communication complexity of computing a function (polynomial\nfor simplicity) and show a lower bound on it being the rank of the matrix of its second\nderivatives, earlier this matrix in the frames of communication complexity was employed in\n[1]. This slightly resembles the lower bound on the bit communication complexity being the\nlogarithm of the rank of the communication matrix [15].\nIn Section 2 we describe the (deterministic) communication protocols (respectively, proba-\nbilistic communication protocols) and relying on this we define the (deterministic) communi-\ncation complexity of recognizing a set (respectively, probabilistic communication complexity).\nAs an application of the matrix of the second derivatives we establish a lower bound n −3\non a probabilistic communication complexity of recognizing a constructible set in C2n whose\nZariski closure contains the hypersurface {f = X1Y1 +· · ·+XnYn = 0}. As a real counterpart\nwe establish the same bound n −3 for a semialgebraic set in R2n whose euclidean closure has\n(full) (2n −1)-dimensional intersection with the hypersurface {f = 0}.\nIn Section 3 we demonstrate a possible exponential gap between the deterministic and\nprobabilistic communication complexities. Namely, we prove a (sharp) lower bound n1 + n2\non the deterministic communication complexity of recognizing the orthant\n{(x1, . . . , xn1, y1, . . . , yn2) ∈Rn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nOn the other hand, we show that the probabilistic communication complexity of recognizing\nthe orthant does not exceed 4.\nIn Section 2 the lower bound was established for a set which involves a polynomial f\nwith a big communication complexity of its computation.\nIn Section 4 we construct sets\ndefined by linear contraints which nevertheless have big probabilistic communication com-\nplexity (clearly, any linear function has the communication complexity of its computation\nat most 2). Namely, we consider the polyhedron {Xi + Yi > 0, 1 ≤i ≤n} ⊂R2n and the\narrangement ∪1≤i,j≤n{Xi + Yj = 0} ⊂R2n and for each of both prove a lower bound n/2\non the probabilistic communication complexity of its recognizing. For the complex arrange-\nment ∪1≤i,j≤n{Xi + Yj = 0} ⊂C2n we establish a lower bound n/4. As applications the\nobtained lower bounds imply the same bounds for the EMPTINESS problem, i. e whether\n{x1, . . . , xn} ∩{y1, . . . , yn} = ∅, and for the KNAPSACK problem.\n1\nLower bound on the communication complexity of comput-\ning a function\nFirst we describe computational models for the communication complexity over com-\nplex\nor\nreal\nnumbers.\nLet\ntwo\nfamilies\nof\nvariables\nX\n=\n{X1, . . . , Xn1}\nand\nY\n=\n{Y1, . . . , Yn2}\nbe\ngiven.\nAs\nusually\nin\ncommunication\ncomplexity\nstudies,\nthere are two parties.\nWe assume that one party is able to calculate polynomials\na1(X), . . . , ar1(X) in X and the second party is able to calculate polynomials b1(Y ), . . . , br2(Y )\n2"},{"paragraph_id":"p3","order":3,"text":"in Y .\nThen the result is obtained by means of calculating suitable polynomials\nP1(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )), . . . , PN(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )). The\ngoal is to minimize r1 + r2 viewed as a measure of communication complexity.\nWe study the communication complexity of two problems:\ncomputing a polynomial\ng(X, Y ) and recognizing a subset S in (n1 + n2)-dimensional complex or real space.\nDefinition 1.1 A polynomial g(X, Y ) has a communication complexity c(g) less or equal\nto r1 + r2 if g\n=\nP(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )) for appropriate polynomials\nP, a1, . . . , ar1, b1, . . . , br2.\nObviously, the communication complexity of g does not exceed n1 + n2.\nBy H(g) denote n1 ×n2 matrix of the second derivatives (\n∂2g\n∂Xi∂Yj ), by H(P) denote r1 ×r2\nmatrix (\n∂2P\n∂ai1∂bj1 ), by the Jacobian J(a1, . . . , ar1) denote n1 × r1 matrix of the first derivatives\n(\n∂ai1\n∂Xi ), similar J(b1, . . . , br2) = (\n∂bj1\n∂Yj ). Then we have\nH(g) = J(a1, . . . , ar1)H(P)(J(b1, . . . , br2))T .\nLemma 1.2 (cf. [1]) In the notations of Definition 1.1 we have\nc(g) ≥min{r1, r2} ≥rk(H(g)).\nCorollary 1.3 c(f = X1Y1 + · · · + XnYn) ≥n\nTo deal in the sequel with communication protocols we need the following statement\ngeneralizing the latter corollary.\nLemma 1.4 Let a polynomial g be a multiple of f. Then rk(H(g)) ≥n −3.\nProof. We write g = f mh where f does not divide h (evidently, f is absolutely irredicible\nwhen n ≥2, we assume here that n1 = n2 = n). We have\nH(g) = mf m−1h"},{"paragraph_id":"p4","order":4,"text":"∂2f\n∂Xi∂Yj"},{"paragraph_id":"p5","order":5,"text":"+ f m"},{"paragraph_id":"p6","order":6,"text":"∂2h\n∂Xi∂Yj"},{"paragraph_id":"p7","order":7,"text":"+\nm(m −1)f m−2h\n ∂f\n∂Xi\n ∂f\n∂Yj"},{"paragraph_id":"p8","order":8,"text":"+ mf m−1\n ∂f\n∂Xi\n ∂h\n∂Yj"},{"paragraph_id":"p9","order":9,"text":"+ mf m−1\n ∂h\n∂Xi\n ∂f\n∂Yj"},{"paragraph_id":"p10","order":10,"text":".\nEach of the latter three matrices has rank at most 1, so it suffices to verify that the sum of\nthe former two matrices divided by f m−1 is non-singular, it equals\nM = mh"},{"paragraph_id":"p11","order":11,"text":"∂2f\n∂Xi∂Yj"},{"paragraph_id":"p12","order":12,"text":"+ f"},{"paragraph_id":"p13","order":13,"text":"∂2h\n∂Xi∂Yj"},{"paragraph_id":"p14","order":14,"text":"We have det(M) = (mh)n + ff1 for a certain polynomial f1, hence det(M) ̸= 0.\nIt would be interesting to clarify, whether one can majorate c(g) via an appropriate\nfunction in rk(H(g))?\n3"},{"paragraph_id":"p15","order":15,"text":"2\nProbabilistic communication protocols\nNow we define a communication protocol for recognizing a set S. We consider two cases:\nS ⊂Cn1+n2 is a constructible set or S ⊂Rn1+n2 is a semialgebraic set.\nA protocol is\na rooted tree, and to its root an input (x, y) = (x1, . . . , xn1, y1, . . . , yn2) is attached.\nTo\nevery vertex v of the tree (including the root, but excluding the leaves) either a certain\npolynomial av(X) or a polynomial bv(Y ) is attached (so, it is calculated either by the first\nparty or by the second party, respectively). To every vertex v (of a depth r) leads a unique\npath from the root, denote by q1(X, Y ), . . . , qr(X, Y ) the polynomials attached to the ver-\ntices v1, . . . , vr = v along this path, thus for every 1 ≤k ≤r either qk(X, Y ) = avk(X)\nor qk(X, Y ) = bvk(Y ), respectively.\nIn addition, to the vertex v a family of testing poly-\nnomials Pv,1(Q1, . . . , Qr), . . . , Pv,Nv(Q1, . . . , Qr) is assigned.\nSimilar to the usual decision\ntrees (see\ne.g.\n[14], [9], [10]) the protocol ramifies at v according to the set of the\nsigns sgn(Pv,1(q1(x, y), . . . , qr(x, y))), . . . , sgn(Pv,Nv(q1(x, y), . . . , qr(x, y))).\nSimilar to deci-\nsion trees in the complex case the sign can attain two values: =, ̸=, in the real case three\nvalues: =, <, >. To every leaf a label either “accept” or “reject” is assigned which provides an\noutput of the protocol. To the protocol naturally corresponds a decision tree (without restric-\ntions on the degrees of testing polynomials). To any input (x, y) corresponds a unique leaf\nof the protocol and a path leading to this leaf, according to the signs of testing polynomials:\nthe output assigned to the leaf is “accept” if and only if (x, y) ∈S.\nThe communication complexity of the recognizing protocol is defined as its depth. We note\nthat the communication complexity counts just the number of the polynomials avi(X) or\nbvi(Y ), respectively, calculated (separately) by each of both parties in several rounds along a\npath of the protocol and ignores the (jointly) calculated polynomials Pv,1, . . . , Pv,Nv.\nNow we introduce probabilistic communication protocols.\nOne can define it similar to\nprobabilistic decision trees (cf. [14], [9], [8], [10]) as a finite family C = {Ci}i of communication\nprotocols Ci, chosen with a certain probability pi ≥0, where P\ni pi = 1. As for decision trees\nwe require that a probabilistic communication protocol for any input returns a correct output\nwith the probability greater than 2/3 (we suppose that a certain continuous probabilistic\nmeasure is fixed in the ambient space,\ne.g.\none can take the Gaussian measure).\nThe\nmaximal depth of communication protocols which constitute a probabilistic communication\nprotocol is called the probabilistic communication complexity.\nFirst we consider probabilistic communication protocols over complex numbers.\nProposition 2.1 The probabilistic communication complexity of an (2n −1)-dimensional\nconstructible set W ⊂C2n such that its Zariski closure W contains the hypersurface U =\n{f = X1Y1 + · · · + XnYn = 0} is greater or equal to n −3.\nProof. Let a probabilistic communication protocol C recognize W. Among communica-\ntion protocols which constitute C there exists C0 such that it gives the correct outputs for at\nleast of 1/3 of the points from U and for at least of 1/3 of the points outside of U (in fact,\nfor the arguments below, instead of 1/3 any positive constant would suffice).\nDistinguish in the decision tree corresponding to C0 a (unique) path along which all the\nsigns in the ramifications are ̸=. Denote by {Pj(q1(x, y), . . . , qr(x, y))}1≤j≤N the collection\nof all the testing polynomials along this path, clearly r does not exceed the communication\ncomplexity of C0. Denote P = Q\n1≤j≤N Pj(q1, . . . , qr). Then the inputs from the Zariski-open\nset V = {(x, y) : P(x, y) ̸= 0} ⊂C2n follow this path in C0.\n4"},{"paragraph_id":"p16","order":16,"text":"Due to the choice of C0 we conclude that f divides P. Indeed, C0 rejects all the points\nfrom a suitable (constructive) subset of C2n of the dimension 2n because C0 rejects a subset\nof a positive (namely, at least 1/3) measure, whence if f did not divide P then C0 would\nreject all the points of U except for its certain (constructive) subset of the dimension at most\n2n −2, but on the other hand, C0 should accept a subset of a positive measure (at least 1/3)\nfrom U. Therefore, Lemma 1.4 and Lemma 1.2 imply that r ≥c(P) ≥rk(H(P)) ≥n −3.\nFor a semialgebraic set S ⊂Rn1+n2 denote by ∂(S) ⊂Rn1+n2 its boundary, being a\nsemialgebraic set as well. The following proposition is a real counterpart of Proposition 2.1.\nCorollary 2.2 The probabilistic communication complexity of a semialgebraic set S such that\ndim(∂(S) ∩U) = 2n −1 is greater or equal to n −3.\nProof. For any communication protocol Ci from C consider the product P (Ci) =\nQ\n1≤j≤N Pj of all the testing polynomials from Ci (cf. the proof of Proposition 2.1 where\na similar product of the polynomials along a particular path was taken).\nFor any point\nu ∈∂(S) ∩U there exists C0 such that P (C0)(u) = 0, otherwise all the points from an appro-\npriate ball centered at u would get the same output for all communication protocols Ci from\nC which would contradict the definition of the boundary. Hence there exists C0 for which f\ndivides P (C0). Therefore, we complete the proof as at the end of Proposition 2.1.\n3\nCommunication complexity of recognizing the orthant\nNow we proceed to estimating the communication complexity of the orthant T\n=\n{(x1, . . . , xn1, y1, . . . , yn2) ∈Rn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nFor this\ngoal we use infinitesimals ǫ1 > · · · > ǫn1+n2 > 0 (see e.g.[7], [9], [8], [10]). Namely, denote by\nRi =\n^\nR(ǫ1, . . . , ǫi) by recursion on i the real closure of the field R(ǫ1, . . . , ǫi), for the base of\nrecursion we put R0 = R. Then ǫi+1 is transcendental over Ri and for any positive element\n0 < d ∈Ri we have 0 < ǫi+1 < d.\nFor a polynomial g ∈R[X1, . . . , Xn1, Y1, . . . , Yn2] denote by lt(g) its least term with respect\nto the following (lexicographical) ordering: take the terms with a minimal degree in Yn2,\namong them with a minimal degree in Yn2−1 and so on. If lt(g) = g0Xi1\n1 · · · X\nin1\nn1 Y j1\n1 · · · Y\njn2\nn2\nfor a certain g0 ∈R, we call (i1, . . . , in1, j1, . . . , jn2) the exponent vector of lt(g).\nTake\ne1, . . . , en1+n2 ∈{−1, 1}, then we have (cf. [9], [8], [10])\nsgn(g(e1ǫ1, . . . , en1+n2ǫn1+n2)) = sgn(lt(g)(e1ǫ1, . . . , en1+n2ǫn1+n2))\n(1)\nLemma 3.1 Let g1, . . . , gs ∈R[X1, . . . , Xn1, Y1, . . . , Yn2] and P1, . . . , PN ∈R[G1, . . . , Gs].\nThen among the exponent vectors of the least terms of P1(g1, . . . , gs), . . . , PN(g1, . . . , gs) there\nare at most s linearly independent.\nProof. We claim that if exponent vectors of any family of polynomials h1, . . . , ht ∈\nR[X1, . . . , Xn1, Y1, . . . , Yn2] are linearly independent then h1, . . . , ht are algebraically indepen-\ndent over R. Indeed, denote the exponent vectors of lt(h1), . . . , lt(ht) by l1, . . . , lt, respectively,\nand denote by L the t × (n1 + n2) matrix with the rows l1, . . . , lt, then for any polynomial\nP = P\nK pKGK ∈R[G1, . . . , Gt] the exponent of the least term of P(h1, . . . , ht) coincides with\nthe least vector among the pairwise distinct vectors KL for all K ∈Zt such that pK ̸= 0.\nThe proved claim entails the lemma immediately.\n5"},{"paragraph_id":"p17","order":17,"text":"Theorem 3.2 The communication complexity of recognizing the orthant T (as well as its\nclosure T in the euclidean topology) is greater or equal to n1 + n2.\nProof. Let a communication protocol C0 recognize T (the arguing for T is similar). Using\nthe Tarski’s transfer principle (see e. g. [7], [9], [8], [10]) one can extend the inputs of C0\nover the field Rn1+n2, then C0 recognizes the set T (Rn1+n2) = {(x1, . . . , xn1, y1, . . . , yn2) ∈\nRn1+n2\nn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nTake in C0 the path which follows\nthe input (ǫ1, . . . , ǫn1+n2) ∈T (Rn1+n2).\nLet r be the length of this path and denote by\nq1(X, Y ), . . . , qr(X, Y ) the polynomials attached to the vertices along this path (we use the\nnotations introduced in Section 2 and recall that every qi depends either on X or on Y ,\nalthough the latter is not used in the proof of the Theorem, cf. Remark 3.3 below). Let\nP1(q1, . . . , qr), . . . , PN(q1, . . . , qr) be all the testing polynomials along this path.\nLemma\n3.1\nimplies\nthat\namong\nthe\nexponent\nvectors\nof\nlt(P1(q1, . . . , qr)), . . . , lt(PN(q1, . . . , qr))\nthere\nare\nat\nmost\nr\nlinearly\nindependent\nK1, . . . , Kr0, r0 ≤r.\nSuppose that the theorem is wrong and r < n1 + n2.\nPick a\nboolean vector 0 ̸= (m1, . . . , mn1+n2) ∈(Z/2Z)n1+n2 orthogonal to all Ki(mod 2), 1 ≤i ≤r0.\nThen\nsgn(Pj(q1, . . . , qr)(ǫ1, . . . , ǫn1+n2)) = sgn(Pj(q1, . . . , qr)((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2))\nfor 1 ≤j ≤N (cf. the proof of lemma 1 [9]). This means that the output of C0 is the same for\nthe inputs (ǫ1, . . . , ǫn1+n2) and ((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2). The obtained contradiction\nwith the supposition completes the proof of the theorem.\nRemark 3.3 The bound in Theorem 3.2 still holds if instead of communication protocols\none considers more general decision trees omitting the condition that each of the polynomials\nq1(X, Y ), . . . , qr(X, Y ) depends either on X or on Y . This strengthens slightly lemma 1 [9]\nsince here we consider decision trees without a priori bound on fan-out of branching, unlike\n[9] where the fan-out did not exceed 3.\nRemark 3.4 Clearly, the communication complexity in the theorem equals n1 + n2.\nRemark 3.5 The probabilistic communication complexity of recognizing the closure T\ndoes not exceed logO(1)(n1 + n2).\nIndeed, the first party tests whether for an input\n(x1, . . . , xn1, y1, . . . , yn2) the inequalities x1 ≥0, . . . , xn1 ≥0 hold by means of a probabilis-\ntic decision tree of the depth logO(1) n1 due to Theorem 1 [9]. The second party tests the\ninequalities y1 ≥0, . . . , yn2 ≥0 by the same token.\nThe latter remark demonstrates an exponential gap between the probabilistic and de-\nterministic communication complexities for recognizing the closure T. The next proposition\nprovides even a bigger gap for T.\nProposition 3.6 The probabilistic communication complexity of recognizing T is at most 4.\nProof. For an input (x1, . . . , xn1, y1, . . . , yn2) consider the partition of the indices\n{1, . . . , n1 + n2} = I0 ∪I+ ∪I−into the subsets for which the corresponding coordinates\nof the input are zero, positive or negative, respectively. If I0 ∪I−̸= ∅then for a randomly\n6"},{"paragraph_id":"p18","order":18,"text":"chosen subset I ⊆{1, . . . , n1+n2} the probability of the event that I ∩I0 = ∅and that |I ∩I−|\nis even is less or equal to 1/2. The latter statement is obvious when I0 ̸= ∅, and when I0 = ∅\nthis probability equals to 1/2.\nTherefore, when (x1, . . . , xn1, y1, . . . , yn2) /∈T and if one chooses randomly a product\nQ\ni1∈I1 xi1\nQ\ni2∈I2 yi2 then this product is positive with the probability less or equal to 1/2.\nThus, the first party chooses randomly independently two subsets I(1), I(2) ⊆{1, . . . , n1} and\ncalculates the products Q\ni∈I(1) xi and Q\ni∈I(2) xi (in a similar way the second party). If all 4\ncalculated products are positive then the output is “accept”, otherwise “reject”.\n4\nLower bound on probabilistic communication complexity\nCorollary 2.2 together with Lemma 1.4 show that if the (n1 +n2−1)-dimensional boundary of\na semialgebraic set contains a “facet” with a great communication complexity of computing\nthe polynomial which determines this facet, then the probabilistic communication complexity\nof recognizing this set is great as well. Now we construct a set (being a polyhedron) with a\ngreat probabilistic communication complexity (note that any facet of the polyhedron being\ndetermined by a linear function, has a communication complexity at most 2).\nConsider the polyhedron S = {(x1, . . . , xn, y1, . . . , yn) ∈R2n : xi + yi > 0, 1 ≤i ≤n}\nand an arrangement R either real (i. e. ⊂R2n) or complex (i. e. ⊂C2n) being a union of\nhyperplanes among which there appear n hyperplanes {Xi + Yi = 0}, 1 ≤i ≤n.\nTheorem 4.1 The probabilistic communication complexity of recognizing over the reals the\nset S or the set R is greater than n/2.\nProof.\nDenote\nZi\n=\nXi + Yi,\n1\n≤\ni\n≤\nn.\nWe\nconsider the new\nco-\nordinates\n(X1, . . . , Xn, Z1, . . . , Zn)\nin\nR2n\nand\nthe\npoint\nu\n=\n(ǫ1, . . . , ǫ2n).\nLet\na\nprobabilistic\ncommunication\nprotocol\nC\nrecognize\nS\n(respectively,\nR).\nIntro-\nduce\nn\npoints\nui\n=\n(ǫ1, . . . , ǫn+i−1, −ǫn+i, ǫn+i+1, . . . , ǫ2n)\n(respectively,\nu(0)\ni\n=\n(ǫ1, . . . , ǫn+i−1, 0, ǫn+i+1, . . . , ǫ2n)), 1 ≤i ≤n.\nClearly, u ∈S, ui\n/∈S (respectively,\nu /∈R, u(0)\ni\n∈R).\nThere exists a communication protocol C0 from the family constituting C which gives\ncorrect outputs for the input u and for at least of n/2 inputs among ui (respectively, u(0)\ni ).\nWithout loss of generality one can assume that the outputs are correct for all ui, 1 ≤i ≤⌈n/2⌉\n(respectively, for u(0)\ni ).\nTake the path in C0 which follows the input u and consider the testing polyno-\nmials P1(q1, . . . , qr), . . . , PN(q1, . . . , qr) along this path (cf.\nSection 2).\nDenote P\n=\nQ\n1≤j≤N Pj(q1, . . . , qr). We claim that the least term lt(P) = Q\n1≤j≤N lt(Pj(q1, . . . , qr)) di-\nvides on each Zi, 1 ≤i ≤⌈n/2⌉(recall that the least term is defined with respect to the\ncoordinates (X1, . . . , Xn, Z1, . . . , Zn)). Otherwise, if lt(P) does not divide on Zi then we have\nsgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(ui)), 1 ≤j ≤N\n(respectively,\nsgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(u(0)\ni ))).\nHence C0 gives the same output for both inputs u and ui (respectively, u(0)\ni ). The obtained\ncontradiction proves the claim.\n7"},{"paragraph_id":"p19","order":19,"text":"Thus, the theorem would follow from the next lemma taking into account Lemma 1.2.\nLemma 4.2 If for a certain k > 1 the product Z1 · · · Zk divides lt(P) then for the rank of\nn × n matrix we have\nrk"},{"paragraph_id":"p20","order":20,"text":"∂2P\n∂Xi∂Yj"},{"paragraph_id":"p21","order":21,"text":"≥k\nProof. Let lt(P) = p0Xm1\n1\n· · · Xmn\nn\nZl1\n1 · · · Zln\nn where p0 ∈R. Then the highest term (cf.\n(1)) of a non-diagonal entry\n∂2P\n∂Xi∂Yj (u) when i ̸= j, 1 ≤i, j ≤k equals\nliljǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫn+iǫn+j\nThe highest term of a diagonal entry\n∂2P\n∂Xi∂Yi (u) either equals\nli(li −1)ǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫ2\nn+i\nwhen li > 1 or is less than\nǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫ2\nn+i\n.\nDenote by M k × k matrix with the diagonal (i, i)-entries li(li −1) and the non-diagonal\n(i, j)-entries lilj, 1 ≤i, j ≤k. Then det(M) = (−1)k+1l1 · · · lk(l1 + · · · + lk −1) ̸= 0 when\nk > 1. Therefore, the coefficient of the k × k minor\ndet"},{"paragraph_id":"p22","order":22,"text":"∂2P\n∂Xi∂Yj"},{"paragraph_id":"p23","order":23,"text":"(u)\nwhere 1 ≤i, j ≤k at its highest term\n(ǫm1\n1\n· · · ǫmn\nn )kǫkl1−2\nn+1 · · · ǫkln−2\n2n\nequals to det(M) and thereby, it does not vanish, which proves the lemma.\nRemark 4.3 The same bound as in the theorem holds as well for the (euclidean) closure S.\nCorollary 4.4 The probabilistic communication complexity over complex numbers of R is\ngreater than n/4.\nProof. Having a probabilistic communication protocol C over C which recognizes R,\none can convert it into a probabilistic communication protocol C(R) over reals which rec-\nognizes R at the cost of increasing the complexity at most twice.\nFor this purpose the\nfirst party replaces every polynomial a(X) in C which the first party calculates by a pair\nof polynomials Re(a), Im(a) ∈R[X] in C(R) where a = Re(a) + √−1Im(a).\nThe same\nfor the second party.\nThen for each testing polynomial Pj(q1, . . . , qr) its real and imagi-\nnary parts Re(Pj(q1, . . . , qr)), Im(Pj(q1, . . . , qr)) can be expressed as polynomials over R in\nRe(ql), Im(ql), 1 ≤l ≤r. Any ramification condition Pj(q1, . . . , qr) = 0 in C we replace in\n8"},{"paragraph_id":"p24","order":24,"text":"C(R) by Re(Pj(q1, . . . , qr)) = Im(Pj(q1, . . . , qr)) = 0. To complete the proof of the corollary\nwe apply Theorem 4.1 to C(R).\nAs particular cases consider the problem EMPTINESS: whether the intersection of two\nfinite sets {x1, . . . , xn} ∩{y1, . . . , yn} = ∅is empty?\nIt corresponds to the arrangement\n∪i,j{xi = yj} (in C2n or R2n).\nAnother example is the KNAPSACK problem: whether\nthere exist subsets I1, I2 ⊆{1, . . . , n} such that P\ni1∈I1 xi1 + P\ni2∈I2 yi2 = 0? It can be also\nrepresented as an arrangement (cf. [10]).\nCorollary 4.5 The probabilistic communication complexity of both EMPTINESS and\nKNAPSACK problems is greater than n/4 over C and greater than n/2 over R.\nAcknowledgements.\nThe author is grateful to the Max-Planck Institut fuer Math-\nematik, Bonn where the paper was written, to Farid Ablayev and to Harry Buhrman for\ninteresting discussions and to anonymous referees for very detailed comments, which helped\nto improve the presentation of the paper.\nReferences\n[1] H. Abelson, Lower bounds on information transfer in distributed computations, J. Assoc.\nComput. Mach., 27 (1980), 384–392.\n[2] F. Ablayev, S. Ablayeva, A discrete approximation and communication complexity ap-\nproach to the superposition problem, in Proc. Intern. Symp. Fundamentals of Computa-\ntion Theory, Lect. Notes Comput. Sci., 2138, (2001), Springer, 47–58.\n[3] M. Bl ̈aser, E. Vicari, Algebraic communication complexity, Preprint (2007).\n[4] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and real computations, Springer\n(1998).\n[5] H. Buhrman, R. de Wolf, Communication complexity lower bounds by polynomials, Proc.\nIEEE Conf. Computational Complexity (2001), 120–130.\n[6] P. B ̈urgisser, Completeness and reduction in algebraic complexity theory, Springer (2000).\n[7] D. Grigoriev, N. Vorobjov, Solving systems of polynomial inequalities in subexponential\ntime, J. Symb. Comput., 5 (1988), 37–64.\n[8] D. Grigoriev, M. Karpinski, F. Meyer auf der Heide, R. Smolensky, A lower bound for\nrandomized algebraic decision trees, Computational Complexity, 6 (1996/1997), 357–375.\n[9] D. Grigoriev, M. Karpinski, R. Smolensky, Randomization and the computational power\nof analytic and algebraic decision trees, Computational Complexity, 6 (1996/1997), 376–\n388.\n[10] D. Grigoriev, Randomized complexity lower bounds for arrangements and polyhedra, Dis-\ncrete Computational Geometry, 21 (1999), 329–344.\n[11] J. Krajiˆcek. Interpolation by a game, Math. Logic Quat., 44 (1998), 450–458.\n9"},{"paragraph_id":"p25","order":25,"text":"[12] E. Kushilevitz, N. Nisan, Communication complexity, Cambridge (1997).\n[13] L. Lovasz, Communication complexity: a survey, in “Paths, flows and VLSI layout”,\nKorte, Lovasz, Proemel, Schrijver Eds. (1990), Springer, 235–266.\n[14] F. Meyer auf der Heide, Simulating probabilistic by deterministic algebraic computation\ntrees, Theor. Comp. Sci., 41 (1984), 325–330.\n[15] A. Yao, Some complexity questions related to distributive computing, in Proc. ACM Symp.\nTheory on Computing (1979), 209–213.\n10"}],"pages":[{"page":1,"text":"arXiv:0710.2732v1 [cs.CC] 15 Oct 2007\nProbabilistic communication complexity over the reals\nDima Grigoriev\nCNRS, IRMAR, Universit ́e de Rennes\nBeaulieu, 35042, Rennes, France\ndmitry.grigoryev@univ-rennes1.fr\nhttp://perso.univ-rennes1.fr/dmitry.grigoryev\nAbstract\nDeterministic and probabilistic communication protocols are introduced in which par-\nties can exchange the values of polynomials (rather than bits in the usual setting). It\nis established a sharp lower bound 2n on the communication complexity of recognizing\nthe 2n-dimensional orthant, on the other hand the probabilistic communication complex-\nity of its recognizing does not exceed 4. A polyhedron and a union of hyperplanes are\nconstructed in R2n for which a lower bound n/2 on the probabilistic communication com-\nplexity of recognizing each is proved. As a consequence this bound holds also for the\nEMPTINESS and the KNAPSACK problems.\nIntroduction\nCommunication complexity (see [15], a survey one can find in [12], [13]) in the usual (bit)\nsetting counts the number of bit exchanges between two (or more) parties who altogether\ncompute a certain function (one of the goals of the communication complexity was to pro-\nvide a framework to analyze distributed computations and to obtain lower bounds on other\ncomplexity ressources). In [2] one can find the relations of the communication complexity\nwith the question of representing a function as a composition of functions of a special form\n(this question stems from the Hilbert’s 13th problem). In [5] the communication complexity\nof quantum computations was studied.\nIn the present paper we introduce the model of communication protocols over real (or com-\nplex) numbers when the parties exchange the values of polynomials. The variables of polyno-\nmials are supposed to be partitioned in two groups: X = {X1, . . . , Xn1}, Y = {Y1, . . . , Yn2},\nthe first party is able to calculate polynomials in X, the second party in Y . It is worthwhile\nto mention that in [11] a different (less restrictive) concept of a communication protocol was\nintroduced in which the parties can exchange arbitrary real numbers (rather than just val-\nues of a given family of polynomials as in the present paper). After the present paper had\nbeen submitted the paper [3] has appeared in which a similar algebraic communication pro-\ntocol was introduced and several lower bounds on the algebraic communication complexity\nfor computing rational functions and recognizing algebraic varieties were established. Unlike\n[3] we obtain lower bounds on probabilistic communication complexity and in addition, for\nrecognizing real semi-algebraic sets.\nWe note that parallel to the numerous customary (boolean or discrete) complexity classes\none develops also their continuous (algebraic or semi-algebraic) counterparts (see e. g. [4],\n[6]). This paper presents an attempt to introduce and study the probabilistic continuous\ncommunication complexity.\n1"},{"page":2,"text":"For illustration of the results obtained in the present paper we consider the KNAP-\nSACK problem: whether for given sets {x1, . . . , xn} and {y1, . . . , yn} there exist subsets\nI1, I2 ⊆{1, . . . , n} such that P\ni1∈I1 xi1 + P\ni2∈I2 yi2 = 0? There is an evident deterministic\ncommunication protocol for the KNAPSACK problem with the communication complexity\n2n when two parties just yield {x1, . . . , xn} and {y1, . . . , yn}, respectively. In Section 4 we\nshow a lower bound n/4 in the complex case and n/2 in the real case on the probabilistic\ncommunication complexity for the KNAPSACK problem.\nIn Section 1 we define the communication complexity of computing a function (polynomial\nfor simplicity) and show a lower bound on it being the rank of the matrix of its second\nderivatives, earlier this matrix in the frames of communication complexity was employed in\n[1]. This slightly resembles the lower bound on the bit communication complexity being the\nlogarithm of the rank of the communication matrix [15].\nIn Section 2 we describe the (deterministic) communication protocols (respectively, proba-\nbilistic communication protocols) and relying on this we define the (deterministic) communi-\ncation complexity of recognizing a set (respectively, probabilistic communication complexity).\nAs an application of the matrix of the second derivatives we establish a lower bound n −3\non a probabilistic communication complexity of recognizing a constructible set in C2n whose\nZariski closure contains the hypersurface {f = X1Y1 +· · ·+XnYn = 0}. As a real counterpart\nwe establish the same bound n −3 for a semialgebraic set in R2n whose euclidean closure has\n(full) (2n −1)-dimensional intersection with the hypersurface {f = 0}.\nIn Section 3 we demonstrate a possible exponential gap between the deterministic and\nprobabilistic communication complexities. Namely, we prove a (sharp) lower bound n1 + n2\non the deterministic communication complexity of recognizing the orthant\n{(x1, . . . , xn1, y1, . . . , yn2) ∈Rn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nOn the other hand, we show that the probabilistic communication complexity of recognizing\nthe orthant does not exceed 4.\nIn Section 2 the lower bound was established for a set which involves a polynomial f\nwith a big communication complexity of its computation.\nIn Section 4 we construct sets\ndefined by linear contraints which nevertheless have big probabilistic communication com-\nplexity (clearly, any linear function has the communication complexity of its computation\nat most 2). Namely, we consider the polyhedron {Xi + Yi > 0, 1 ≤i ≤n} ⊂R2n and the\narrangement ∪1≤i,j≤n{Xi + Yj = 0} ⊂R2n and for each of both prove a lower bound n/2\non the probabilistic communication complexity of its recognizing. For the complex arrange-\nment ∪1≤i,j≤n{Xi + Yj = 0} ⊂C2n we establish a lower bound n/4. As applications the\nobtained lower bounds imply the same bounds for the EMPTINESS problem, i. e whether\n{x1, . . . , xn} ∩{y1, . . . , yn} = ∅, and for the KNAPSACK problem.\n1\nLower bound on the communication complexity of comput-\ning a function\nFirst we describe computational models for the communication complexity over com-\nplex\nor\nreal\nnumbers.\nLet\ntwo\nfamilies\nof\nvariables\nX\n=\n{X1, . . . , Xn1}\nand\nY\n=\n{Y1, . . . , Yn2}\nbe\ngiven.\nAs\nusually\nin\ncommunication\ncomplexity\nstudies,\nthere are two parties.\nWe assume that one party is able to calculate polynomials\na1(X), . . . , ar1(X) in X and the second party is able to calculate polynomials b1(Y ), . . . , br2(Y )\n2"},{"page":3,"text":"in Y .\nThen the result is obtained by means of calculating suitable polynomials\nP1(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )), . . . , PN(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )). The\ngoal is to minimize r1 + r2 viewed as a measure of communication complexity.\nWe study the communication complexity of two problems:\ncomputing a polynomial\ng(X, Y ) and recognizing a subset S in (n1 + n2)-dimensional complex or real space.\nDefinition 1.1 A polynomial g(X, Y ) has a communication complexity c(g) less or equal\nto r1 + r2 if g\n=\nP(a1(X), . . . , ar1(X), b1(Y ), . . . , br2(Y )) for appropriate polynomials\nP, a1, . . . , ar1, b1, . . . , br2.\nObviously, the communication complexity of g does not exceed n1 + n2.\nBy H(g) denote n1 ×n2 matrix of the second derivatives (\n∂2g\n∂Xi∂Yj ), by H(P) denote r1 ×r2\nmatrix (\n∂2P\n∂ai1∂bj1 ), by the Jacobian J(a1, . . . , ar1) denote n1 × r1 matrix of the first derivatives\n(\n∂ai1\n∂Xi ), similar J(b1, . . . , br2) = (\n∂bj1\n∂Yj ). Then we have\nH(g) = J(a1, . . . , ar1)H(P)(J(b1, . . . , br2))T .\nLemma 1.2 (cf. [1]) In the notations of Definition 1.1 we have\nc(g) ≥min{r1, r2} ≥rk(H(g)).\nCorollary 1.3 c(f = X1Y1 + · · · + XnYn) ≥n\nTo deal in the sequel with communication protocols we need the following statement\ngeneralizing the latter corollary.\nLemma 1.4 Let a polynomial g be a multiple of f. Then rk(H(g)) ≥n −3.\nProof. We write g = f mh where f does not divide h (evidently, f is absolutely irredicible\nwhen n ≥2, we assume here that n1 = n2 = n). We have\nH(g) = mf m−1h\n \n∂2f\n∂Xi∂Yj\n \n+ f m\n \n∂2h\n∂Xi∂Yj\n \n+\nm(m −1)f m−2h\n ∂f\n∂Xi\n ∂f\n∂Yj\n \n+ mf m−1\n ∂f\n∂Xi\n ∂h\n∂Yj\n \n+ mf m−1\n ∂h\n∂Xi\n ∂f\n∂Yj\n \n.\nEach of the latter three matrices has rank at most 1, so it suffices to verify that the sum of\nthe former two matrices divided by f m−1 is non-singular, it equals\nM = mh\n \n∂2f\n∂Xi∂Yj\n \n+ f\n \n∂2h\n∂Xi∂Yj\n \nWe have det(M) = (mh)n + ff1 for a certain polynomial f1, hence det(M) ̸= 0.\nIt would be interesting to clarify, whether one can majorate c(g) via an appropriate\nfunction in rk(H(g))?\n3"},{"page":4,"text":"2\nProbabilistic communication protocols\nNow we define a communication protocol for recognizing a set S. We consider two cases:\nS ⊂Cn1+n2 is a constructible set or S ⊂Rn1+n2 is a semialgebraic set.\nA protocol is\na rooted tree, and to its root an input (x, y) = (x1, . . . , xn1, y1, . . . , yn2) is attached.\nTo\nevery vertex v of the tree (including the root, but excluding the leaves) either a certain\npolynomial av(X) or a polynomial bv(Y ) is attached (so, it is calculated either by the first\nparty or by the second party, respectively). To every vertex v (of a depth r) leads a unique\npath from the root, denote by q1(X, Y ), . . . , qr(X, Y ) the polynomials attached to the ver-\ntices v1, . . . , vr = v along this path, thus for every 1 ≤k ≤r either qk(X, Y ) = avk(X)\nor qk(X, Y ) = bvk(Y ), respectively.\nIn addition, to the vertex v a family of testing poly-\nnomials Pv,1(Q1, . . . , Qr), . . . , Pv,Nv(Q1, . . . , Qr) is assigned.\nSimilar to the usual decision\ntrees (see\ne.g.\n[14], [9], [10]) the protocol ramifies at v according to the set of the\nsigns sgn(Pv,1(q1(x, y), . . . , qr(x, y))), . . . , sgn(Pv,Nv(q1(x, y), . . . , qr(x, y))).\nSimilar to deci-\nsion trees in the complex case the sign can attain two values: =, ̸=, in the real case three\nvalues: =, <, >. To every leaf a label either “accept” or “reject” is assigned which provides an\noutput of the protocol. To the protocol naturally corresponds a decision tree (without restric-\ntions on the degrees of testing polynomials). To any input (x, y) corresponds a unique leaf\nof the protocol and a path leading to this leaf, according to the signs of testing polynomials:\nthe output assigned to the leaf is “accept” if and only if (x, y) ∈S.\nThe communication complexity of the recognizing protocol is defined as its depth. We note\nthat the communication complexity counts just the number of the polynomials avi(X) or\nbvi(Y ), respectively, calculated (separately) by each of both parties in several rounds along a\npath of the protocol and ignores the (jointly) calculated polynomials Pv,1, . . . , Pv,Nv.\nNow we introduce probabilistic communication protocols.\nOne can define it similar to\nprobabilistic decision trees (cf. [14], [9], [8], [10]) as a finite family C = {Ci}i of communication\nprotocols Ci, chosen with a certain probability pi ≥0, where P\ni pi = 1. As for decision trees\nwe require that a probabilistic communication protocol for any input returns a correct output\nwith the probability greater than 2/3 (we suppose that a certain continuous probabilistic\nmeasure is fixed in the ambient space,\ne.g.\none can take the Gaussian measure).\nThe\nmaximal depth of communication protocols which constitute a probabilistic communication\nprotocol is called the probabilistic communication complexity.\nFirst we consider probabilistic communication protocols over complex numbers.\nProposition 2.1 The probabilistic communication complexity of an (2n −1)-dimensional\nconstructible set W ⊂C2n such that its Zariski closure W contains the hypersurface U =\n{f = X1Y1 + · · · + XnYn = 0} is greater or equal to n −3.\nProof. Let a probabilistic communication protocol C recognize W. Among communica-\ntion protocols which constitute C there exists C0 such that it gives the correct outputs for at\nleast of 1/3 of the points from U and for at least of 1/3 of the points outside of U (in fact,\nfor the arguments below, instead of 1/3 any positive constant would suffice).\nDistinguish in the decision tree corresponding to C0 a (unique) path along which all the\nsigns in the ramifications are ̸=. Denote by {Pj(q1(x, y), . . . , qr(x, y))}1≤j≤N the collection\nof all the testing polynomials along this path, clearly r does not exceed the communication\ncomplexity of C0. Denote P = Q\n1≤j≤N Pj(q1, . . . , qr). Then the inputs from the Zariski-open\nset V = {(x, y) : P(x, y) ̸= 0} ⊂C2n follow this path in C0.\n4"},{"page":5,"text":"Due to the choice of C0 we conclude that f divides P. Indeed, C0 rejects all the points\nfrom a suitable (constructive) subset of C2n of the dimension 2n because C0 rejects a subset\nof a positive (namely, at least 1/3) measure, whence if f did not divide P then C0 would\nreject all the points of U except for its certain (constructive) subset of the dimension at most\n2n −2, but on the other hand, C0 should accept a subset of a positive measure (at least 1/3)\nfrom U. Therefore, Lemma 1.4 and Lemma 1.2 imply that r ≥c(P) ≥rk(H(P)) ≥n −3.\nFor a semialgebraic set S ⊂Rn1+n2 denote by ∂(S) ⊂Rn1+n2 its boundary, being a\nsemialgebraic set as well. The following proposition is a real counterpart of Proposition 2.1.\nCorollary 2.2 The probabilistic communication complexity of a semialgebraic set S such that\ndim(∂(S) ∩U) = 2n −1 is greater or equal to n −3.\nProof. For any communication protocol Ci from C consider the product P (Ci) =\nQ\n1≤j≤N Pj of all the testing polynomials from Ci (cf. the proof of Proposition 2.1 where\na similar product of the polynomials along a particular path was taken).\nFor any point\nu ∈∂(S) ∩U there exists C0 such that P (C0)(u) = 0, otherwise all the points from an appro-\npriate ball centered at u would get the same output for all communication protocols Ci from\nC which would contradict the definition of the boundary. Hence there exists C0 for which f\ndivides P (C0). Therefore, we complete the proof as at the end of Proposition 2.1.\n3\nCommunication complexity of recognizing the orthant\nNow we proceed to estimating the communication complexity of the orthant T\n=\n{(x1, . . . , xn1, y1, . . . , yn2) ∈Rn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nFor this\ngoal we use infinitesimals ǫ1 > · · · > ǫn1+n2 > 0 (see e.g.[7], [9], [8], [10]). Namely, denote by\nRi =\n^\nR(ǫ1, . . . , ǫi) by recursion on i the real closure of the field R(ǫ1, . . . , ǫi), for the base of\nrecursion we put R0 = R. Then ǫi+1 is transcendental over Ri and for any positive element\n0 < d ∈Ri we have 0 < ǫi+1 < d.\nFor a polynomial g ∈R[X1, . . . , Xn1, Y1, . . . , Yn2] denote by lt(g) its least term with respect\nto the following (lexicographical) ordering: take the terms with a minimal degree in Yn2,\namong them with a minimal degree in Yn2−1 and so on. If lt(g) = g0Xi1\n1 · · · X\nin1\nn1 Y j1\n1 · · · Y\njn2\nn2\nfor a certain g0 ∈R, we call (i1, . . . , in1, j1, . . . , jn2) the exponent vector of lt(g).\nTake\ne1, . . . , en1+n2 ∈{−1, 1}, then we have (cf. [9], [8], [10])\nsgn(g(e1ǫ1, . . . , en1+n2ǫn1+n2)) = sgn(lt(g)(e1ǫ1, . . . , en1+n2ǫn1+n2))\n(1)\nLemma 3.1 Let g1, . . . , gs ∈R[X1, . . . , Xn1, Y1, . . . , Yn2] and P1, . . . , PN ∈R[G1, . . . , Gs].\nThen among the exponent vectors of the least terms of P1(g1, . . . , gs), . . . , PN(g1, . . . , gs) there\nare at most s linearly independent.\nProof. We claim that if exponent vectors of any family of polynomials h1, . . . , ht ∈\nR[X1, . . . , Xn1, Y1, . . . , Yn2] are linearly independent then h1, . . . , ht are algebraically indepen-\ndent over R. Indeed, denote the exponent vectors of lt(h1), . . . , lt(ht) by l1, . . . , lt, respectively,\nand denote by L the t × (n1 + n2) matrix with the rows l1, . . . , lt, then for any polynomial\nP = P\nK pKGK ∈R[G1, . . . , Gt] the exponent of the least term of P(h1, . . . , ht) coincides with\nthe least vector among the pairwise distinct vectors KL for all K ∈Zt such that pK ̸= 0.\nThe proved claim entails the lemma immediately.\n5"},{"page":6,"text":"Theorem 3.2 The communication complexity of recognizing the orthant T (as well as its\nclosure T in the euclidean topology) is greater or equal to n1 + n2.\nProof. Let a communication protocol C0 recognize T (the arguing for T is similar). Using\nthe Tarski’s transfer principle (see e. g. [7], [9], [8], [10]) one can extend the inputs of C0\nover the field Rn1+n2, then C0 recognizes the set T (Rn1+n2) = {(x1, . . . , xn1, y1, . . . , yn2) ∈\nRn1+n2\nn1+n2 : xi > 0, yj > 0, 1 ≤i ≤n1, 1 ≤j ≤n2}.\nTake in C0 the path which follows\nthe input (ǫ1, . . . , ǫn1+n2) ∈T (Rn1+n2).\nLet r be the length of this path and denote by\nq1(X, Y ), . . . , qr(X, Y ) the polynomials attached to the vertices along this path (we use the\nnotations introduced in Section 2 and recall that every qi depends either on X or on Y ,\nalthough the latter is not used in the proof of the Theorem, cf. Remark 3.3 below). Let\nP1(q1, . . . , qr), . . . , PN(q1, . . . , qr) be all the testing polynomials along this path.\nLemma\n3.1\nimplies\nthat\namong\nthe\nexponent\nvectors\nof\nlt(P1(q1, . . . , qr)), . . . , lt(PN(q1, . . . , qr))\nthere\nare\nat\nmost\nr\nlinearly\nindependent\nK1, . . . , Kr0, r0 ≤r.\nSuppose that the theorem is wrong and r < n1 + n2.\nPick a\nboolean vector 0 ̸= (m1, . . . , mn1+n2) ∈(Z/2Z)n1+n2 orthogonal to all Ki(mod 2), 1 ≤i ≤r0.\nThen\nsgn(Pj(q1, . . . , qr)(ǫ1, . . . , ǫn1+n2)) = sgn(Pj(q1, . . . , qr)((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2))\nfor 1 ≤j ≤N (cf. the proof of lemma 1 [9]). This means that the output of C0 is the same for\nthe inputs (ǫ1, . . . , ǫn1+n2) and ((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2). The obtained contradiction\nwith the supposition completes the proof of the theorem.\nRemark 3.3 The bound in Theorem 3.2 still holds if instead of communication protocols\none considers more general decision trees omitting the condition that each of the polynomials\nq1(X, Y ), . . . , qr(X, Y ) depends either on X or on Y . This strengthens slightly lemma 1 [9]\nsince here we consider decision trees without a priori bound on fan-out of branching, unlike\n[9] where the fan-out did not exceed 3.\nRemark 3.4 Clearly, the communication complexity in the theorem equals n1 + n2.\nRemark 3.5 The probabilistic communication complexity of recognizing the closure T\ndoes not exceed logO(1)(n1 + n2).\nIndeed, the first party tests whether for an input\n(x1, . . . , xn1, y1, . . . , yn2) the inequalities x1 ≥0, . . . , xn1 ≥0 hold by means of a probabilis-\ntic decision tree of the depth logO(1) n1 due to Theorem 1 [9]. The second party tests the\ninequalities y1 ≥0, . . . , yn2 ≥0 by the same token.\nThe latter remark demonstrates an exponential gap between the probabilistic and de-\nterministic communication complexities for recognizing the closure T. The next proposition\nprovides even a bigger gap for T.\nProposition 3.6 The probabilistic communication complexity of recognizing T is at most 4.\nProof. For an input (x1, . . . , xn1, y1, . . . , yn2) consider the partition of the indices\n{1, . . . , n1 + n2} = I0 ∪I+ ∪I−into the subsets for which the corresponding coordinates\nof the input are zero, positive or negative, respectively. If I0 ∪I−̸= ∅then for a randomly\n6"},{"page":7,"text":"chosen subset I ⊆{1, . . . , n1+n2} the probability of the event that I ∩I0 = ∅and that |I ∩I−|\nis even is less or equal to 1/2. The latter statement is obvious when I0 ̸= ∅, and when I0 = ∅\nthis probability equals to 1/2.\nTherefore, when (x1, . . . , xn1, y1, . . . , yn2) /∈T and if one chooses randomly a product\nQ\ni1∈I1 xi1\nQ\ni2∈I2 yi2 then this product is positive with the probability less or equal to 1/2.\nThus, the first party chooses randomly independently two subsets I(1), I(2) ⊆{1, . . . , n1} and\ncalculates the products Q\ni∈I(1) xi and Q\ni∈I(2) xi (in a similar way the second party). If all 4\ncalculated products are positive then the output is “accept”, otherwise “reject”.\n4\nLower bound on probabilistic communication complexity\nCorollary 2.2 together with Lemma 1.4 show that if the (n1 +n2−1)-dimensional boundary of\na semialgebraic set contains a “facet” with a great communication complexity of computing\nthe polynomial which determines this facet, then the probabilistic communication complexity\nof recognizing this set is great as well. Now we construct a set (being a polyhedron) with a\ngreat probabilistic communication complexity (note that any facet of the polyhedron being\ndetermined by a linear function, has a communication complexity at most 2).\nConsider the polyhedron S = {(x1, . . . , xn, y1, . . . , yn) ∈R2n : xi + yi > 0, 1 ≤i ≤n}\nand an arrangement R either real (i. e. ⊂R2n) or complex (i. e. ⊂C2n) being a union of\nhyperplanes among which there appear n hyperplanes {Xi + Yi = 0}, 1 ≤i ≤n.\nTheorem 4.1 The probabilistic communication complexity of recognizing over the reals the\nset S or the set R is greater than n/2.\nProof.\nDenote\nZi\n=\nXi + Yi,\n1\n≤\ni\n≤\nn.\nWe\nconsider the new\nco-\nordinates\n(X1, . . . , Xn, Z1, . . . , Zn)\nin\nR2n\nand\nthe\npoint\nu\n=\n(ǫ1, . . . , ǫ2n).\nLet\na\nprobabilistic\ncommunication\nprotocol\nC\nrecognize\nS\n(respectively,\nR).\nIntro-\nduce\nn\npoints\nui\n=\n(ǫ1, . . . , ǫn+i−1, −ǫn+i, ǫn+i+1, . . . , ǫ2n)\n(respectively,\nu(0)\ni\n=\n(ǫ1, . . . , ǫn+i−1, 0, ǫn+i+1, . . . , ǫ2n)), 1 ≤i ≤n.\nClearly, u ∈S, ui\n/∈S (respectively,\nu /∈R, u(0)\ni\n∈R).\nThere exists a communication protocol C0 from the family constituting C which gives\ncorrect outputs for the input u and for at least of n/2 inputs among ui (respectively, u(0)\ni ).\nWithout loss of generality one can assume that the outputs are correct for all ui, 1 ≤i ≤⌈n/2⌉\n(respectively, for u(0)\ni ).\nTake the path in C0 which follows the input u and consider the testing polyno-\nmials P1(q1, . . . , qr), . . . , PN(q1, . . . , qr) along this path (cf.\nSection 2).\nDenote P\n=\nQ\n1≤j≤N Pj(q1, . . . , qr). We claim that the least term lt(P) = Q\n1≤j≤N lt(Pj(q1, . . . , qr)) di-\nvides on each Zi, 1 ≤i ≤⌈n/2⌉(recall that the least term is defined with respect to the\ncoordinates (X1, . . . , Xn, Z1, . . . , Zn)). Otherwise, if lt(P) does not divide on Zi then we have\nsgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(ui)), 1 ≤j ≤N\n(respectively,\nsgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(u(0)\ni ))).\nHence C0 gives the same output for both inputs u and ui (respectively, u(0)\ni ). The obtained\ncontradiction proves the claim.\n7"},{"page":8,"text":"Thus, the theorem would follow from the next lemma taking into account Lemma 1.2.\nLemma 4.2 If for a certain k > 1 the product Z1 · · · Zk divides lt(P) then for the rank of\nn × n matrix we have\nrk\n \n∂2P\n∂Xi∂Yj\n \n≥k\nProof. Let lt(P) = p0Xm1\n1\n· · · Xmn\nn\nZl1\n1 · · · Zln\nn where p0 ∈R. Then the highest term (cf.\n(1)) of a non-diagonal entry\n∂2P\n∂Xi∂Yj (u) when i ̸= j, 1 ≤i, j ≤k equals\nliljǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫn+iǫn+j\nThe highest term of a diagonal entry\n∂2P\n∂Xi∂Yi (u) either equals\nli(li −1)ǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫ2\nn+i\nwhen li > 1 or is less than\nǫm1\n1\n· · · ǫmn\nn ǫl1\nn+1 · · · ǫln\n2n\nǫ2\nn+i\n.\nDenote by M k × k matrix with the diagonal (i, i)-entries li(li −1) and the non-diagonal\n(i, j)-entries lilj, 1 ≤i, j ≤k. Then det(M) = (−1)k+1l1 · · · lk(l1 + · · · + lk −1) ̸= 0 when\nk > 1. Therefore, the coefficient of the k × k minor\ndet\n \n∂2P\n∂Xi∂Yj\n \n(u)\nwhere 1 ≤i, j ≤k at its highest term\n(ǫm1\n1\n· · · ǫmn\nn )kǫkl1−2\nn+1 · · · ǫkln−2\n2n\nequals to det(M) and thereby, it does not vanish, which proves the lemma.\nRemark 4.3 The same bound as in the theorem holds as well for the (euclidean) closure S.\nCorollary 4.4 The probabilistic communication complexity over complex numbers of R is\ngreater than n/4.\nProof. Having a probabilistic communication protocol C over C which recognizes R,\none can convert it into a probabilistic communication protocol C(R) over reals which rec-\nognizes R at the cost of increasing the complexity at most twice.\nFor this purpose the\nfirst party replaces every polynomial a(X) in C which the first party calculates by a pair\nof polynomials Re(a), Im(a) ∈R[X] in C(R) where a = Re(a) + √−1Im(a).\nThe same\nfor the second party.\nThen for each testing polynomial Pj(q1, . . . , qr) its real and imagi-\nnary parts Re(Pj(q1, . . . , qr)), Im(Pj(q1, . . . , qr)) can be expressed as polynomials over R in\nRe(ql), Im(ql), 1 ≤l ≤r. Any ramification condition Pj(q1, . . . , qr) = 0 in C we replace in\n8"},{"page":9,"text":"C(R) by Re(Pj(q1, . . . , qr)) = Im(Pj(q1, . . . , qr)) = 0. To complete the proof of the corollary\nwe apply Theorem 4.1 to C(R).\nAs particular cases consider the problem EMPTINESS: whether the intersection of two\nfinite sets {x1, . . . , xn} ∩{y1, . . . , yn} = ∅is empty?\nIt corresponds to the arrangement\n∪i,j{xi = yj} (in C2n or R2n).\nAnother example is the KNAPSACK problem: whether\nthere exist subsets I1, I2 ⊆{1, . . . , n} such that P\ni1∈I1 xi1 + P\ni2∈I2 yi2 = 0? It can be also\nrepresented as an arrangement (cf. [10]).\nCorollary 4.5 The probabilistic communication complexity of both EMPTINESS and\nKNAPSACK problems is greater than n/4 over C and greater than n/2 over R.\nAcknowledgements.\nThe author is grateful to the Max-Planck Institut fuer Math-\nematik, Bonn where the paper was written, to Farid Ablayev and to Harry Buhrman for\ninteresting discussions and to anonymous referees for very detailed comments, which helped\nto improve the presentation of the paper.\nReferences\n[1] H. Abelson, Lower bounds on information transfer in distributed computations, J. Assoc.\nComput. Mach., 27 (1980), 384–392.\n[2] F. Ablayev, S. Ablayeva, A discrete approximation and communication complexity ap-\nproach to the superposition problem, in Proc. Intern. Symp. Fundamentals of Computa-\ntion Theory, Lect. Notes Comput. Sci., 2138, (2001), Springer, 47–58.\n[3] M. Bl ̈aser, E. Vicari, Algebraic communication complexity, Preprint (2007).\n[4] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and real computations, Springer\n(1998).\n[5] H. Buhrman, R. de Wolf, Communication complexity lower bounds by polynomials, Proc.\nIEEE Conf. Computational Complexity (2001), 120–130.\n[6] P. B ̈urgisser, Completeness and reduction in algebraic complexity theory, Springer (2000).\n[7] D. Grigoriev, N. Vorobjov, Solving systems of polynomial inequalities in subexponential\ntime, J. Symb. Comput., 5 (1988), 37–64.\n[8] D. Grigoriev, M. Karpinski, F. Meyer auf der Heide, R. Smolensky, A lower bound for\nrandomized algebraic decision trees, Computational Complexity, 6 (1996/1997), 357–375.\n[9] D. Grigoriev, M. Karpinski, R. Smolensky, Randomization and the computational power\nof analytic and algebraic decision trees, Computational Complexity, 6 (1996/1997), 376–\n388.\n[10] D. Grigoriev, Randomized complexity lower bounds for arrangements and polyhedra, Dis-\ncrete Computational Geometry, 21 (1999), 329–344.\n[11] J. Krajiˆcek. Interpolation by a game, Math. Logic Quat., 44 (1998), 450–458.\n9"},{"page":10,"text":"[12] E. Kushilevitz, N. Nisan, Communication complexity, Cambridge (1997).\n[13] L. Lovasz, Communication complexity: a survey, in “Paths, flows and VLSI layout”,\nKorte, Lovasz, Proemel, Schrijver Eds. (1990), Springer, 235–266.\n[14] F. Meyer auf der Heide, Simulating probabilistic by deterministic algebraic computation\ntrees, Theor. Comp. Sci., 41 (1984), 325–330.\n[15] A. Yao, Some complexity questions related to distributive computing, in Proc. ACM Symp.\nTheory on Computing (1979), 209–213.\n10"}]},"section_tree":[],"assets":{"figures":[],"tables":[],"images":[]},"math_expressions":{"inline":[],"block":[{"equation_id":"eq1","equation_number":null,"raw_text":"mials are supposed to be partitioned in two groups: X = {X1, . . . , Xn1}, Y = {Y1, . . . , Yn2},","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq2","equation_number":null,"raw_text":"i2∈I2 yi2 = 0? There is an evident deterministic","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq3","equation_number":null,"raw_text":"Zariski closure contains the hypersurface {f = X1Y1 +· · ·+XnYn = 0}. As a real counterpart","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq4","equation_number":null,"raw_text":"(full) (2n −1)-dimensional intersection with the hypersurface {f = 0}.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq5","equation_number":null,"raw_text":"arrangement ∪1≤i,j≤n{Xi + Yj = 0} ⊂R2n and for each of both prove a lower bound n/2","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq6","equation_number":null,"raw_text":"ment ∪1≤i,j≤n{Xi + Yj = 0} ⊂C2n we establish a lower bound n/4. As applications the","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq7","equation_number":null,"raw_text":"{x1, . . . , xn} ∩{y1, . . . , yn} = ∅, and for the KNAPSACK problem.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq8","equation_number":null,"raw_text":"∂Xi ), similar J(b1, . . . , br2) = (","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq9","equation_number":null,"raw_text":"H(g) = J(a1, . . . , ar1)H(P)(J(b1, . . . , br2))T .","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq10","equation_number":null,"raw_text":"Corollary 1.3 c(f = X1Y1 + · · · + XnYn) ≥n","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq11","equation_number":null,"raw_text":"Proof. We write g = f mh where f does not divide h (evidently, f is absolutely irredicible","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq12","equation_number":null,"raw_text":"when n ≥2, we assume here that n1 = n2 = n). We have","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq13","equation_number":null,"raw_text":"H(g) = mf m−1h","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq14","equation_number":null,"raw_text":"M = mh","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq15","equation_number":null,"raw_text":"We have det(M) = (mh)n + ff1 for a certain polynomial f1, hence det(M) ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq16","equation_number":null,"raw_text":"a rooted tree, and to its root an input (x, y) = (x1, . . . , xn1, y1, . . . , yn2) is attached.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq17","equation_number":null,"raw_text":"tices v1, . . . , vr = v along this path, thus for every 1 ≤k ≤r either qk(X, Y ) = avk(X)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq18","equation_number":null,"raw_text":"or qk(X, Y ) = bvk(Y ), respectively.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq19","equation_number":null,"raw_text":"sion trees in the complex case the sign can attain two values: =, ̸=, in the real case three","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq20","equation_number":null,"raw_text":"values: =, <, >. To every leaf a label either “accept” or “reject” is assigned which provides an","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq21","equation_number":null,"raw_text":"probabilistic decision trees (cf. [14], [9], [8], [10]) as a finite family C = {Ci}i of communication","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq22","equation_number":null,"raw_text":"i pi = 1. As for decision trees","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq23","equation_number":null,"raw_text":"constructible set W ⊂C2n such that its Zariski closure W contains the hypersurface U =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq24","equation_number":null,"raw_text":"{f = X1Y1 + · · · + XnYn = 0} is greater or equal to n −3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq25","equation_number":null,"raw_text":"signs in the ramifications are ̸=. Denote by {Pj(q1(x, y), . . . , qr(x, y))}1≤j≤N the collection","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq26","equation_number":null,"raw_text":"complexity of C0. Denote P = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq27","equation_number":null,"raw_text":"set V = {(x, y) : P(x, y) ̸= 0} ⊂C2n follow this path in C0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq28","equation_number":null,"raw_text":"dim(∂(S) ∩U) = 2n −1 is greater or equal to n −3.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq29","equation_number":null,"raw_text":"Proof. For any communication protocol Ci from C consider the product P (Ci) =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq30","equation_number":null,"raw_text":"u ∈∂(S) ∩U there exists C0 such that P (C0)(u) = 0, otherwise all the points from an appro-","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq31","equation_number":null,"raw_text":"Ri =","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq32","equation_number":null,"raw_text":"recursion we put R0 = R. Then ǫi+1 is transcendental over Ri and for any positive element","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq33","equation_number":null,"raw_text":"among them with a minimal degree in Yn2−1 and so on. If lt(g) = g0Xi1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq34","equation_number":null,"raw_text":"sgn(g(e1ǫ1, . . . , en1+n2ǫn1+n2)) = sgn(lt(g)(e1ǫ1, . . . , en1+n2ǫn1+n2))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq35","equation_number":null,"raw_text":"P = P","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq36","equation_number":null,"raw_text":"the least vector among the pairwise distinct vectors KL for all K ∈Zt such that pK ̸= 0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq37","equation_number":null,"raw_text":"over the field Rn1+n2, then C0 recognizes the set T (Rn1+n2) = {(x1, . . . , xn1, y1, . . . , yn2) ∈","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq38","equation_number":null,"raw_text":"boolean vector 0 ̸= (m1, . . . , mn1+n2) ∈(Z/2Z)n1+n2 orthogonal to all Ki(mod 2), 1 ≤i ≤r0.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq39","equation_number":null,"raw_text":"sgn(Pj(q1, . . . , qr)(ǫ1, . . . , ǫn1+n2)) = sgn(Pj(q1, . . . , qr)((−1)m1ǫ1, . . . , (−1)mn1+n2ǫn1+n2))","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq40","equation_number":null,"raw_text":"{1, . . . , n1 + n2} = I0 ∪I+ ∪I−into the subsets for which the corresponding coordinates","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq41","equation_number":null,"raw_text":"of the input are zero, positive or negative, respectively. If I0 ∪I−̸= ∅then for a randomly","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq42","equation_number":null,"raw_text":"chosen subset I ⊆{1, . . . , n1+n2} the probability of the event that I ∩I0 = ∅and that |I ∩I−|","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq43","equation_number":null,"raw_text":"is even is less or equal to 1/2. The latter statement is obvious when I0 ̸= ∅, and when I0 = ∅","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq44","equation_number":null,"raw_text":"Consider the polyhedron S = {(x1, . . . , xn, y1, . . . , yn) ∈R2n : xi + yi > 0, 1 ≤i ≤n}","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq45","equation_number":null,"raw_text":"hyperplanes among which there appear n hyperplanes {Xi + Yi = 0}, 1 ≤i ≤n.","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq46","equation_number":null,"raw_text":"1≤j≤N Pj(q1, . . . , qr). We claim that the least term lt(P) = Q","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq47","equation_number":null,"raw_text":"sgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(ui)), 1 ≤j ≤N","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq48","equation_number":null,"raw_text":"sgn(Pj(q1, . . . , qr)(u)) = sgn(Pj(q1, . . . , qr)(u(0)","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq49","equation_number":null,"raw_text":"Proof. Let lt(P) = p0Xm1","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq50","equation_number":null,"raw_text":"∂Xi∂Yj (u) when i ̸= j, 1 ≤i, j ≤k equals","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq51","equation_number":null,"raw_text":"(i, j)-entries lilj, 1 ≤i, j ≤k. Then det(M) = (−1)k+1l1 · · · lk(l1 + · · · + lk −1) ̸= 0 when","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq52","equation_number":null,"raw_text":"of polynomials Re(a), Im(a) ∈R[X] in C(R) where a = Re(a) + √−1Im(a).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq53","equation_number":null,"raw_text":"Re(ql), Im(ql), 1 ≤l ≤r. Any ramification condition Pj(q1, . . . , qr) = 0 in C we replace in","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq54","equation_number":null,"raw_text":"C(R) by Re(Pj(q1, . . . , qr)) = Im(Pj(q1, . . . , qr)) = 0. To complete the proof of the corollary","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq55","equation_number":null,"raw_text":"finite sets {x1, . . . , xn} ∩{y1, . . . , yn} = ∅is empty?","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq56","equation_number":null,"raw_text":"∪i,j{xi = yj} (in C2n or R2n).","raw_latex":null,"normalized_latex":null,"display_type":"block_candidate"},{"equation_id":"eq57","equation_number":null,"raw_text":"i2∈I2 yi2 = 0? It can be also","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":27096,"parse_confidence":0.5,"equation_parse_rate_proxy":1.0,"citation_resolution_rate_proxy":0.0,"structure_coverage_proxy":0.0,"asset_coverage_proxy":0.0,"accepted_for_training":true}}