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values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
813efa7375df91f8f92bd7da792f00c70cd32d83 | subsection | 98 | 122 | Concluding the ESSP and the SSP for | Note, that for the sake of simplicity we sometimes present a set S \subseteq S(U^{\sigma _5}_\varphi )\cup S(U^{\sigma _6}_\varphi ) such that sup^{\sigma _5}=S\cap S(U^{\sigma _5}_\varphi ) and sup^{\sigma _6}=S\cap S(U^{\sigma _6}_\varphi ).
Such a presented support sup^\sigma allows always a signature sig^\sigma tha... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.024427982047200203,
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0.030195487663149834,
0.06786736845970154,
0.031583961099386215,
0.02339044027030468,
0.01850789599120617,
0.025938518345355988,
-0.01588352769613266,
0.01... | |
aae6cc451a267659f501f89a88061f3aa4404461 | subsection | 99 | 122 | Concluding the ESSP and the SSP for | Therefore, for the sake of simplicity, in the sequel we often refer to a given support sup^\sigma as to the region (sup^\sigma ,sig^\sigma ) which it allows and, e.g., say sup^\sigma inhibits e at s instead of (sup^\sigma ,sig^\sigma ) inhibits e at s.In the following proofs, for n\in \lbrace 0,\dots , m-1\rbrace if we... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.01912406086921692,
-0.003588623134419322,
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0.02848004549741745,
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0.022436048835515976,
-0.04954245686531067,
... | |
3a528e32eb187b502e6a811f8db4f85f308ed267 | subsection | 100 | 122 | Concluding the ESSP and the SSP for | Furthermore, with respect to T^\sigma _\varphi a key union inhibits k at all states of F^\sigma _T and at t_{i,\alpha _i,2} (t_{i,\beta _i,11}).Hence, by the arbitrariness of i and \alpha _i and the symmetry of the translators, to prove k to be inhibitable in U^\sigma _\varphi it is sufficient to present regions that s... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.0423065721988678,
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-0.023992039263248444,
-0.02814333327114582,
0.0... | |
c31c2f1da99d071079f54f6525eca20e88c6fd81 | subsection | 101 | 122 | Concluding the ESSP and the SSP for | To attack these challenge we define the following sets of states that will help us to compose corresponding supports of U^\sigma _\varphi :S^{\sigma _5}_0=\emptyset , S^{\sigma _6}_0=\lbrace t^{\prime }_{n,0,0},t^{\prime }_{n,0,1}, t^{\prime }_{n,1,0},t^{\prime }_{n,1,1}, t^{\prime }_{n,2,0},t^{\prime }_{n,2,1}\mid n\i... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.01130945049226284,
0.008188285864889622,
-0.03336364030838013,
-0.024938788264989853,
0.013461451046168804,
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0.003600022289901972,
0.03430991247296333,
0.016559721902012825,
0.035836152732372284,
-0.03141005337238312,
0.03519513085484505,
-0.034493058919906616,
0.02... | |
f7d3df52d68cfbda0d8922bf230f3b374ffb4084 | subsection | 102 | 122 | Concluding the ESSP and the SSP for | For \sigma =\sigma _5 (\sigma =\sigma _6) the event v_{3i+\alpha _i} (v_{3i+\beta _i}) is, besides of t^{\prime }_{i,\alpha _i,0} (t^{\prime }_{i,\beta _i,0}), already inhibited in T^{\sigma _5}_{i,\alpha _i} (T^{\sigma _6}_{i,\beta _i}) by using the supports R^k_2,R^k_3 of Lemma REF .
Hence, for v_{3i+\alpha _i} (v_{3... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.037026625126600266,
-0.018513312563300133,
-0.017383892089128494,
0.01284332387149334,
0.00517014367505908,
0.00795171968638897,
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0.06813143193721771,
0.03147110342979431,
0.014621395617723465,
-0.03247842192649841,
-0.0017256069695577025,
-0.050182826817035675,
0.0... | |
dee89cb22554fc9148e06dabde40e279c3b4e948 | subsection | 103 | 122 | Concluding the ESSP and the SSP for | To do so, we define the following sets to be used for composing fitting supports:S_0=\lbrace t^{\prime }_{n,0,0},t^{\prime }_{n,1,0},t^{\prime }_{n,2,0},b^{\prime }_{n,0},b^{\prime }_{n,5}\mid n\in \lbrace 0,\dots ,m-1\rbrace \rbrace
S_1=\lbrace h^{\prime }_{n,0},h^{\prime }_{n,4},h^{\prime }_{n,5},g_{n,0},g_{n,2},g_... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.006400848273187876,
0.027190255001187325,
-0.04641568660736084,
0.031004823744297028,
0.018432002514600754,
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0.005847735796123743,
0.045378122478723526,
0.03918326273560524,
0.03176773712038994,
-0.04498140886425972,
-0.0007610067259520292,
-0.0035532719921320677,
0... | |
c2ab51472f713f51995a3a4c00b39510322fb5aa | subsection | 104 | 122 | Concluding the ESSP and the SSP for | Finally, for \sigma _5 (\sigma _6) the support S_6 (S_7) can be used for the inhibition of v_{3i+\alpha _i} (v_{3i+\beta _i}) at the last state standing h^{\prime }_{3i+\alpha _i,1} (h^{\prime }_{3i+\beta _i,1}).We now argue that w_{3i+\alpha _i} is for \sigma _5 and \sigma _6 inhibitable in T_{i,\alpha _i} and G_{3i+\... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.03311983868479729,
0.03339456766843796,
-0.009645962156355381,
-0.015766264870762825,
0.0018086179625242949,
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0.04932871833443642,
0.05375487357378006,
0.009401760995388031,
-0.03968275710940361,
-0.001357417437247932,
-0.021794991567730904,
0... | |
f80f49f2a5cb759ec60897a98f975fb807f93c7c | subsection | 105 | 122 | Concluding the ESSP and the SSP for | The inhibition at g_{3i+\alpha _i,0}, g_{3i+\alpha _i,2},\dots , g_{3i+\alpha _i,4}, g_{3i+\alpha _i,7}, g_{3i+\alpha _i,8} is done for \sigma _5 (\sigma _6 ) with S_4 (S_5).
Finally, we can complete the set \lbrace g_{3i+\alpha _i,0},g_{3i+\alpha _i,1}, g_{3i+\alpha _i,2},g_{3i+\alpha _i,3},\rbrace , respectively the ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.04591319337487221,
-0.005681910552084446,
-0.005578880198299885,
-0.03037475235760212,
0.002652066992595792,
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0.024437174201011658,
0.04206674173474312,
0.01772115007042885,
-0.03736552596092224,
0.014821047894656658,
-0.028955228626728058,
0.0... | |
cf60f26b3d9fa3423e73663b556769843941459d | subsection | 106 | 122 | Concluding the ESSP and the SSP for | We need the following sets:S^{\sigma _5}_0=\lbrace f^{\prime }_{2,4}, f^{\prime }_{2,5},g_{n,3}, d_{n^{\prime },3}\mid n\in \lbrace 0,\dots , 3m-1\rbrace ,n^{\prime }\in \lbrace 0,\dots , 18m-1\rbrace \rbrace ,
S^{\sigma _5}_1=M_0\cup M_1\cup M_2, where
M_0=\lbrace t^{\prime }_{n,0,0},t^{\prime }_{n,1,0},t^{\prime }_... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.011835593730211258,
-0.02467847242951393,
-0.03952828794717789,
-0.0322331078350544,
0.012369759380817413,
-0.012743676081299782,
-0.02791399136185646,
0.025685757398605347,
-0.006181064527481794,
0.011805069632828236,
-0.05417970195412636,
0.02135138027369976,
-0.03327091410756111,
0.0... | |
66327574b5a30fa6fd5eeccd608b5e881fee06ab | subsection | 107 | 122 | Concluding the ESSP and the SSP for | Let n\in \lbrace 0,\dots , 3m-1\rbrace and n^{\prime }\in \lbrace 0,\dots , 18m-1\rbrace and \sigma \in \lbrace \sigma _5,\sigma _6\rbrace .
The inhibition of y_n and p_n at g_{n,3} and d_{n^{\prime },3} is allowed by the support S^{\sigma }_0.
The support S^{\sigma }_1 allows the inhibition of y_n at g_{n,0} and p_n a... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.03627513721585274,
-0.02610040456056595,
-0.015895161777734756,
-0.05119401589035988,
0.005430592689663172,
-0.040973518043756485,
-0.030585220083594322,
0.025703787803649902,
0.030539456754922867,
0.02236305922269821,
-0.04210234805941582,
0.005792886484414339,
-0.044298991560935974,
-0... | |
f81dca275a973fb72f6b82fda474f771e1e64e76 | subsection | 108 | 122 | Concluding the ESSP and the SSP for | Using the following sets:S_0=\lbrace t^{\prime }_{n,\alpha _n,3}, t^{\prime }_{n,\alpha _n,4}, t^{\prime }_{n,\beta _n,9}, t^{\prime }_{n,\beta _n,10}, t^{\prime }_{n,\gamma _n,6}, t^{\prime }_{n,\gamma _n,7}\mid n\in \lbrace i,j,\ell \rbrace \rbrace
S_1=\lbrace d_{18n+6\alpha _n+3,n^{\prime }}, d_{18n+6\beta _n+2,n^... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.039649561047554016,
0.0018886193865910172,
-0.03250714763998985,
-0.01109516154974699,
-0.02074047550559044,
-0.04984429106116295,
-0.026738272979855537,
0.07227879762649536,
0.051095739006996155,
0.021701954305171967,
-0.028127074241638184,
0.010622053407132626,
-0.030721541494131088,
-... | |
d71691b37e563460299243b046501bc4944ead4d | subsection | 109 | 122 | Concluding the ESSP and the SSP for | Using the setsS_0=\lbrace f^{\prime }_{0,0}, f^{\prime }_{0,3}, f^{\prime }_{0,6},, f^{\prime }_{0,8},f^{\prime }_{1,0},f^{\prime }_{1,4}, f^{\prime }_{2,1}, f^{\prime }_{2,2}, f^{\prime }_{2,5},f^{\prime }_{2,6}, f^{\prime }_{2,7}\rbrace
S_1=\lbrace f^{\prime }_{0,0}, f^{\prime }_{0,3}, f^{\prime }_{0,7},f^{\prime }... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.03621533513069153,
-0.007021106779575348,
-0.03911378234624863,
-0.009358933195471764,
-0.006193523760885,
-0.03224903717637062,
-0.026879282668232918,
0.03877817094326019,
0.021387487649917603,
0.023553695529699326,
-0.05699262395501137,
-0.007730463519692421,
0.02271467261016369,
0.040... | |
3204cb2df8b5cbcc51009e1dc2b45aa8d2ecaef3 | subsection | 110 | 122 | Concluding the ESSP and the SSP for | Finally, S_3=\lbrace h^{\prime }_{n,3}, h^{\prime }_{n,4}, g_{n,3}, g_{n,4}\mid n\in \lbrace 0,\dots , 3m-1\rbrace \rbrace \cup \lbrace t^{\prime }_{n,n^{\prime },0}, t^{\prime }_{n,n^{\prime },1}\mid n\in \lbrace 0,\dots , m-1\rbrace , n^{\prime }\in \lbrace 0,1,2\rbrace \rbracerespectively S(U^{\sigma _6}_\varphi )\s... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.03175826743245125,
0.005796417593955994,
-0.004347313195466995,
-0.06955701112747192,
-0.020119670778512955,
-0.030339669436216354,
-0.02138572931289673,
0.020211191847920418,
0.0036723355296999216,
0.013575820252299309,
-0.016016416251659393,
0.025443222373723984,
-0.009411551989614964,
... | |
6f5fc777d17c986ba1eef6daa375d3e9109d0a7e | subsection | 111 | 122 | Concluding the ESSP and the SSP for | The needed sets are:S_0=\lbrace b^{\prime }_{n,2}, b^{\prime }_{n,3}\mid n\in \lbrace 0,\dots , m-1\rbrace \rbrace
S_1=\lbrace h^{\prime }_{n,0}, h^{\prime }_{n,1}\mid n\in \lbrace 0,\dots , 3m-1\rbrace \rbrace
S_2=\lbrace f^{\prime }_{0,0}, f^{\prime }_{0,1}, f^{\prime }_{0,5}, f^{\prime }_{1,2}, f^{\prime }_{2,0}... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
-0.004413900896906853,
-0.01873905025422573,
-0.04138461500406265,
-0.03274755924940109,
0.0029031794983893633,
-0.05136452987790108,
-0.015229293145239353,
0.025758564472198486,
-0.00775961484760046,
0.01535900216549635,
-0.05621715262532234,
0.012825872749090195,
-0.02050918899476528,
-0... | |
447af54044baf88d374aa9016021d2ffd8dcd1ca | subsection | 112 | 122 | Concluding the ESSP and the SSP for | The remaining states are f^{\prime }_{0,4}, f^{\prime }_{2,4}.
For \sigma _5 the set S_4\cup S_5\cup S_6 is a support of U(F^{\sigma _5}_K,D^{\sigma _5}, G^{\sigma _5}), firstly, allowing the inhibition of q_0 at f^{\prime }_{0,4}, f^{\prime }_{2,4} and, secondly, assuring that k is the only border crossing event in U^... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.028308149427175522,
0.016099782660603523,
-0.0017358771292492747,
-0.038639478385448456,
-0.007836245000362396,
-0.04538460075855255,
-0.005657814908772707,
0.059454746544361115,
0.017091713845729828,
0.04456053674221039,
-0.03097872994840145,
-0.003528981003910303,
-0.014566107653081417,
... | |
edb05d473fd573755b90223f6266cf56c32e5b79 | subsection | 113 | 122 | Concluding the ESSP and the SSP for | The set S_2, respectively S_3, is a support of F^{\sigma _5}_K, firstly, allowing the inhibition of q_1 at f^{\prime }_{0,1}, f^{\prime }_{0,7}, f^{\prime }_{0,8}, respectively at f^{\prime }_{0,0}, and, secondly, assuring that k is the only border crossing event that occurs in U^{\sigma _5}_\varphi \setminus F^{\sigma... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.043945860117673874,
-0.009788635186851025,
-0.0288699883967638,
-0.030105965211987495,
-0.014290033839643002,
-0.05075136572122574,
0.002315550111234188,
0.053528498858213425,
0.028549550101161003,
0.024750065058469772,
-0.04122976213693619,
0.01763937994837761,
-0.010460030287504196,
0.... | |
a3acefd51c1cddd6bd4c102d6321999f6b5359ec | subsection | 114 | 122 | Concluding the ESSP and the SSP for | The following sets are needed:S_0=\lbrace b^{\prime }_{n,3}, b^{\prime }_{n,4}, b^{\prime }_{n,5}, b^{\prime }_{n,6}, b_{n,2}, b_{n,3}, b_{n,4} \mid n\in \lbrace 0,\dots , m-1\rbrace \rbrace ,
S_1=\lbrace b^{\prime }_{n,4}, b^{\prime }_{n,5}, b^{\prime }_{n,6} \mid n\in \lbrace 0,\dots , m-1\rbrace \rbrace ,
S_2=\lbr... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.025701457634568214,
-0.002083283383399248,
-0.022969532757997513,
-0.02719714865088463,
-0.012949933297932148,
-0.053875382989645004,
-0.00035031401785090566,
0.040536265820264816,
0.01979500614106655,
0.03124161809682846,
-0.050426140427589417,
-0.0006963355117477477,
0.002363725332543254... | |
ce0b69ec708f38d852ff7550600d99be25efc058 | subsection | 115 | 122 | Concluding the ESSP and the SSP for | But they are necessary for proving the SSP of U^{\sigma _6}_\varphi in Lemma REF .The events a_0,\dots , a_{18m-1} are inhibitable.
WithS^{\sigma _5}_0=\lbrace d_{n,0}, d_{n,2}, d_{n,3}, d_{n,4}, d_{n,7}, d_{n,8}\mid n\in \lbrace 0,\dots , 18m-1\rbrace \rbrace
S^{\sigma _6}_0= \lbrace d_{n,1}, d_{n,5}, d_{n,6} \mid... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.059400130063295364,
0.001191398361697793,
-0.022191107273101807,
-0.05698871612548828,
-0.016071001067757607,
-0.023076308891177177,
-0.042031850665807724,
0.01314067654311657,
0.030463170260190964,
0.01308725867420435,
-0.03873523324728012,
0.03696483001112938,
-0.03104313090443611,
0.0... | |
79ef3d92901ec712efa0c9259ac5aa742b3b2d7e | subsection | 116 | 122 | Concluding the ESSP and the SSP for | To do so, we prove for i\in \lbrace 0,\dots , m-1\rbrace and \alpha _i\in \lbrace 0,1,2\rbrace , both arbitrary but fixed, that the events a_{18i+6\alpha _i},\dots ,a_{18i+6\alpha _i+5} are inhibitable in T^\sigma _{i,\alpha _i} by regions of U^\sigma _\varphi .Firstly, all a_{18i+6\alpha _i},\dots ,a_{18i+6\alpha _i+5... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
0.09392563998699188,
-0.01218317449092865,
-0.012083999812602997,
-0.024244287982583046,
-0.008109401911497116,
-0.03252915292978287,
-0.013213060796260834,
0.034573666751384735,
0.053767696022987366,
-0.0075868298299610615,
-0.03814394026994705,
-0.010901537723839283,
-0.05297430232167244,
... | |
cb03d7e1066094e3f75027489ce46107b07bbcf2 | subsection | 117 | 122 | Concluding the ESSP and the SSP for | To tackle the schedule, we need the following sets of states:M_0=\lbrace t^{\prime }_{i,\alpha _i,2}, t^{\prime }_{i,\alpha _i,3},t^{\prime }_{i,\alpha _i,4}, t^{\prime }_{i,\alpha _i,5}, t^{\prime }_{i,\alpha _i,8}, t^{\prime }_{i,\alpha _i,11}\rbrace ,
M_1=\lbrace t^{\prime }_{i,\alpha _i,2}, t^{\prime }_{i,\alpha _... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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... | |
6a894612bb02e0d508162babf49ccb9e072b414f | subsection | 118 | 122 | Concluding the ESSP and the SSP for | If M\subseteq S(T^{\sigma _6}_\varphi ) then Acc(M)=\lbrace n\in \lbrace 0,\dots , 18m-1\rbrace \mid \exists s [baseline=-1pt]{
[->,line width=0.3pt] (0,0) -- ++(0.6,0) node[anchor=base, yshift=2pt, pos=0.5] {\scalebox {0.75}{a_n}};
}s^{\prime }\in T^{\sigma _6}_\varphi : s\in M, s^{\prime }\notin M \rbrace .For n\in \... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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0.026... | |
288d4ad7175df06b701bbc5266b3ff59b87c8f52 | subsection | 119 | 122 | Concluding the ESSP and the SSP for | We need the following sets:N_0=\lbrace t^{\prime }_{n,\alpha _n,2},\dots , t^{\prime }_{n,\alpha _n,6}, t^{\prime }_{n,\alpha _n,12},\dots , t^{\prime }_{n,\alpha _n,16}\mid n\in \lbrace i,j,\ell \rbrace \rbrace ,
N_1=\lbrace t^{\prime }_{n,\beta _n,2},t^{\prime }_{n,\beta _n,3}, t^{\prime }_{n,\beta _n,12},t^{\prime ... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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9c02df376e82533033fdc6234ae403866694fa0c | subsection | 120 | 122 | Concluding the ESSP and the SSP for | S^{\sigma _6}_4=\lbrace g_{n,6},g_{n,7},g_{n,8}\mid n\in \lbrace 3n^{\prime },3n^{\prime }+1,3n^{\prime }+2\rbrace , n^{\prime }\in \lbrace i,j,\ell \rbrace \rbrace .
S^{\sigma _5}_5=\lbrace g_{n,6},g_{n,7},g_{n,8}\mid n\in \lbrace 3n^{\prime },3n^{\prime }+1,3n^{\prime }+2\rbrace , n^{\prime }\in \lbrace i,j,\ell \rb... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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-0.... | |
df65cb6de2bc7e907a57c8e0836353f5ee208dfa | subsection | 121 | 122 | Concluding the ESSP and the SSP for | Hence, by the arbitrariness of i and \alpha _i and the symmetry of the construction, we have proven that the events a_{18i+6\alpha _i},\dots ,a_{18i+6\alpha _i+5} are inhibitable at the relevant states of t^{\prime }_{i,\alpha _i,2}, t^{\prime }_{i,\alpha _i,5}, t^{\prime }_{i,\alpha _i,8} and t^{\prime }_{i,\alpha _i,... | {
"cite_spans": []
} | 1806.03703 | Towards Completely Characterizing the Complexity of Boolean Nets
Synthesis | [
"Ronny Tredup",
"Christian Rosenke"
] | [
"cs.CC"
] | 2,018 | en | Computer Science | [
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... | |
b9322df4bcade7e410f23a7a671dbb276ee11543 | abstract | 0 | 80 | Abstract | The study of the stratification associated to the number of generators of the
ideals in the punctual Hilbert scheme of points on the affine plain goes back
to the '70s. In this paper, we present an elegant formula for the E-polynomials
of these strata. | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... | |
d70addb922cca16a35639493a0f3fc4bc5061a61 | subsection | 1 | 80 | Introduction | The study of the topology of the Hilbert scheme of points on the
affine plane has brought a wealth of results in several branches of
mathematics, such as geometric
representation theory, theory of symmetric polynomials, singularities, symplectic geometry and in the
enumerative geometry in two dimension
(, , , , , , , ,... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/ulect/018",
"end": 323,
"openalex_id": "https://openalex.org/W2214822119",
"raw": "L. Göttsche, Hilbert schemes of points on surfaces, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 483–494,... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.011321221478283405,
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0.009841223247349262,
0.04076860100030899,
0.017... | |
f34e7bda08df35d1635590ba57dacc645220ce17 | subsection | 2 | 80 | Introduction | This function is a classical
invariant studied by A. Iarrobino in \cite {Ia} (See §2).An important set of examples of ideals in H^{[n]} are the ideals
generated by monomials. In this case, those monomials that are not
contained in I form a basis\left\lbrace (p,q)\in \mathbb {N}^2 \;\vrule \; x^py^q\notin I \right\rbrac... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s00222-008-0142-x",
"end": 1142,
"openalex_id": "https://openalex.org/W2114120297",
"raw": "T. Hausel and F. Rodriguez-Villegas, Mixed Hodge polynomials of character varieties, Invent. Math. 174 (2008), no. 3, 555–624.",
"sour... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.008401121012866497,
... | |
a6d1bcc08497c79bc0bb3e8290944d016c2ef38e | subsection | 3 | 80 | Introduction | Then Poincaré duality implies the equality of mixed Hodge numbers:h^{p,q;j}_c(Z) = h^{d-p,d-q;2d-j}(Z)where h^{p,q;j}:=\dim _\mathbb {C} \mathrm {gr}_p F\left( \mathrm {gr}_{p+q}W (H^{j}(Z))\right) are the mixed Hodge numbers of Z. Equivalently, if H(Z; u,v,s):=\sum _{p,q,j} h^{p,q;j}(Z) u^p v^q s^j is the mixed Hodge ... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.0030175407882779837,
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-0.0... | |
7f832bd5f6e1cc072ec3fd69c37d99bc045041bd | subsection | 4 | 80 | Introduction | Applying the factorization
property, we have E(n;u,v)=(uv)n, and
the E(n0;u,v)=(uv)n-1.
In particular, the E-polynomial of * is uv-1.
Notation:
Throughout the article, t will be a variable of degree 2.
If the mixed Hodge numbers h^{p,q;j}(Z) of an algebraic variety Z
vanish except when p=q, then the E-polynomial E(Z;u... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0073491",
"end": 876,
"openalex_id": "https://openalex.org/W579869728",
"raw": "L. Göttsche, Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.",... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.012250049039721489,
-0.021555814892053604,
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0.03926727920770645,
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0.023950904607772827,
0.018123360350728035,
-0.0014511655317619443,
0.042318351566791534,
-0.0025533647276461124,... | |
a168184eb30f292261ec98174a2aa8e3d056b097 | subsection | 5 | 80 | Introduction | Cheah , )
The generating function of the E-polynomials of a smooth surface S has the form\sum _{n=0}^\infty E\left(S^{[n]};u,v\right)s^n=
\prod _{d=1}^\infty \prod _{p,q} \left(\frac{1}{1-u^{p+d-1}v^{q+d-1}s^d}\right)^{e_{p,q}(S)},where e_{p,g}:=\sum _k (-1)^k h^{p,q,k}_c(Z).In particular, for the case of \mathbb {C}^... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 573,
"openalex_id": "",
"raw": "I. Grojnowski, Instantons and affine algebras. I. The Hilbert scheme and vertex operators, Math. Res. Lett. 3 (1996), no. 2, 275–291.",
"source_ref_id": "323d658b39a404420be921f019b28c76cedc81... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.038606636226177216,
0.005985554773360491,
-0.05469019338488579,
-0.015274799428880215,
0.0032998756505548954,
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0.010109445080161095,
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0.046541597694158554,
-0.007412321865558624,... | |
c69f9449a48c13d64106efa5ab0840f23eecf8fc | subsection | 6 | 80 | Introduction | \end{array}\right.}Remark 4 Note that the summand indexed by a=1 on the right hand
side of (REF ) does not depend on q,
and thus may be omitted when calculating
E\left(B^{[n]}_m ;t\right) for n>0.Example 5 We list some examples of the generating function from Theorem REF :Case of m=2:
\sum _{n=0}^{\infty } E\left(B^{[... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.05602368712425232,
0.025051986798644066,
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0.02120722457766533,
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0.03237534314393997,
-0.012014884501695633,
-... | |
5cf1249ccc7675e550679d38ea30c43483a767bc | subsection | 7 | 80 | Introduction | Case of m=4:
\sum _{n=0}^{\infty } E\left(B^{[n]}_4; t\right) q^n
=&
\frac{t^3}{(1-t) \left(1-t^2\right) \left(1-t^3\right)}
-\frac{t^4}{(1-t) \left(1-t^2\right)^2}\cdot \prod _{d=0}^{\infty } \frac{ 1- t^{d-2} q^d }{ 1- t^{d-1}q^d }
\\&+\frac{t^6}{(1-t) \left(1-t^2\right) \left(1-t^3\right)}\cdot \prod _{d=0}^{\infty... | {
"cite_spans": [
{
"arxiv_id": "",
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"raw": "H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula, J. Algebraic Geom. 20 (20... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.011734409257769585,
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0.0424514003098011,
0.0... | |
29f7406c21c7b994b8d3e12e994de81381933f90 | subsection | 8 | 80 | Introduction | Then in Section 3, we compute a formula for the generating function of E\left(H^{[n]}_m;t\right) as an application of Theorem REF .
In Section 4, we give a formula of the Euler characteristics of B^{[n]}_m.
We list a table of examples of E\left(B^{[n]}_m;t\right) in Appendix A.Acknowledgments. We would like to thank Al... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.038358382880687714,
-0.00... | |
51426b434eb77f6b65a48f36856a694932fe515a | subsection | 9 | 80 | The refined Hilbert schemes | Definition 6 Let n, r\in \mathbb {N}, r\ge 1.
The refined Hilbert scheme
H^{[n,n+r]}\subset H^{[n]}\times H^{[n,n+r]}
is defined asH^{[n,n+r]}=\lbrace (I,J)\in H^{[n]}\times H^{[n,n+r]}\;\vrule \; I\supset J\supseteq \mathfrak {m}I \rbrace .Here, the condition I\supset J\supseteq \mathfrak {m}I is equivalent to the req... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.015793146565556526,
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0.035065360367298126,
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-0.021286414936184883,
-0.009857364930212498,
... | |
2262cadd8aba5ad4643118fd6f68e6f95993fd6e | subsection | 10 | 80 | The refined Hilbert schemes | Moreover, the quotient
I/J corresponds to a subset S_{I/J} of \Delta _J of r elbows and \Delta _I=\Delta _J\setminus S_{I/J}.
Therefore, we may represent a T-fixed point (I,J) by a pair (\Delta _J, S_{I/J}) of a Young diagram \Delta _J with n+r boxes, and a subset S_{I/J} of r marked elbows of \Delta _J and we denote t... | {
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{
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"raw": "H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999."... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... | |
d7647b482e42e6e4f999472bcb9fc81cc32489b0 | subsection | 11 | 80 | The refined Hilbert schemes | Now, if I\in U_\lambda and G_I is a reduced Gröbner basis of I, then by definition, \mu \left(I_\lambda \right)=|G_I|. Since I is an intersection of ideals with I_0 and \mu (I)=\mu (I_0), we have \mu (I)\le |G_I|= \mu \left(I_\lambda \right) by the Buchberger Algorithm construction of the reduced Gröbner basis.
We conc... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.008469847962260246,
... | |
6c8364ac4d1b5eab4260efe2994b4583e1f0df19 | subsection | 12 | 80 | The refined Hilbert schemes | Moreover, this
*-action on H^{[n]} induces, at the same time, a cell decomposition of the Briançon variety
B^{[n]}=
\mathop {\vphantom{\bigcup }
\displaylimits _{p\in \Pi (n)}\mathbb {A}^{2n-\alpha (p)}, where \alpha (p) is
the number of positive weights of T_p H^{[n]} (See \cite {ES,HR2,HNlecture}).
The same argument... | {
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"raw": "H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula, J. Algebraic Geom. 20 (2... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.... | |
1fb3dcab5de903c66920f2ccf3dc110077822856 | subsection | 13 | 80 | The refined Hilbert schemes | We note that \Delta _I=\Delta _J\backslash S_{I/J} is the Young diagram corresponding to I.
The character of the tangent space T_{(I,J)}H^{[n,n+r]} as a T-module is given by\sum _{\square }\left( T_1^{-l_\Delta ( \square )} T_2^{a_{\Delta _I}( \square )+1} + T_1^{l_{\Delta _I}( \square )+1} T_2^{-a_{\Delta }( \square )... | {
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"arxiv_id": "",
"doi": "10.1090/s1056-3911-10-00534-5",
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"raw": "H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula, J. Algebraic Geom. 20 (2... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.0... | |
531694446243b8f8614824014cc68b5c4df76a36 | subsection | 14 | 80 | The refined Hilbert schemes | Then
\mu (I) must be at least 2, and the only ideal I\in H^{[1]} of codimension 1 with \mu (I)\ge 2 is the
maximal ideal \langle x,y\rangle =\mathfrak {m}. Moreover, since
\dim _{\mathfrak {m}/\mathfrak {m}^2=2, the only possible
J\in H^{[3]}, J\supseteq \mathfrak {m}I is \mathfrak {m}^2. Thus H^{[1,3]} is a point \lef... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0.026827413588762283,
0.03... | |
c1eacbfa7a25bc2f4a4bc449c28b63dd9e0f433b | subsection | 15 | 80 | The refined Hilbert schemes | We recall that if I_\lambda \in (H^{[n]})^T and \Delta _\lambda is the corresponding Young diagram with n boxes, then \mu (I_\lambda ) is equal to the number of "elbows" of \Delta _\lambda .We claim that\max \left\lbrace \mu (I) \;\vrule \; I\in H^{[n]}\right\rbrace =\max \left\lbrace \mu (I) \;\vrule \; I\in \left(H^{... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/ulect/018",
"end": 666,
"openalex_id": "https://openalex.org/W2214822119",
"raw": "H. Nakajima, Lectures on Hilbert schemes of points on surfaces, University Lecture Series, 18, American Mathematical Society, Providence, RI, 1999.",... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.026839669793844223,
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0.017242085188627243,
... | |
8f9c10f944089ed4fd7b7db4334ffc066424a1b9 | subsection | 16 | 80 | The refined Hilbert schemes | It follows that
if I is an ideal with \dim _Ȑ/I=\binom{k}{2} and \mu (I)=k, then it can only be \mathfrak {m}^{k-1}=\langle y^{k-1},y^{k-2}x,\dots ,yx^{k-2},x^{k-1}\rangle , which is the monomial ideal corresponding to the partition \lambda =k-1\ge k-2\ge \cdots \ge 1.
An immediate consequence of Proposition REF is th... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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-0.012737225741147995,
... | |
8cfbe1b1ca84067bf937ad9a5da4dc4576c64966 | subsection | 17 | 80 | The refined Hilbert schemes | The same arguments go through for
H^{[n,n+r]} and B^{[n,n+r]}:
\begin{} We have
\begin{equation*}
{E}\left(H^{[n,n+r]}; t\right)=\sum _{p\in \Pi (n,r)}
t^{\alpha (p)} \text{~~ and~~~}
{E}\left(B^{[n,n+r]}; t\right)=\sum _{p\in \Pi (n,r)}
t^{2n-r(r-1)-\alpha (p)},
\end{equation*}
where \alpha (p) is
the number of positi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s1056-3911-10-00534-5",
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"raw": "H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula, J. Algebraic Geom. 20 (2... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... | |
60b8c3083accda1d7d7cc3890c6940ed9788550a | subsection | 18 | 80 | The refined Hilbert schemes | The character of the tangent space T_{(I,J)}H^{[n,n+r]} as a T-module is given by\sum _{\square }\left( T_1^{-l_\Delta ( \square )} T_2^{a_{\Delta _I}( \square )+1} + T_1^{l_{\Delta _I}( \square )+1} T_2^{-a_{\Delta }( \square )} \right)where the summation runs over all box \square of \Delta _J\backslash \left(S_{I/J}\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/s1056-3911-10-00534-5",
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"openalex_id": "https://openalex.org/W2040328335",
"raw": "H. Nakajima and K. Yoshioka, Perverse coherent sheaves on blow-up. II. Wall-crossing and Betti numbers formula, J. Algebraic Geom. 20 (20... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
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0... | |
c04a9556e95c6544cbb02671f8f9af5af36b1661 | subsection | 19 | 80 | Proof of | We consider the refined Hilbert scheme strataH^{[n]}_m&:=\left\lbrace I\in H^{[n]}\;\vrule \; \mu (I)=m\right\rbrace ,\\
H^{[n]}_m(s)&:=\left\lbrace I\in H^{[n]}~\vrule ~ \mu (I)=m, \sigma (I)=s \right\rbraceand it induces decomposition of H^{[n]}H^{[n]}=
\mathop {\vphantom{\bigcup }
\displaylimits _m H^{[n]}_m.
}IfH[... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.02832798659801483,
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-0.0004910107818432152,
0.01501825638115406,
... | |
bde8d3be7fcb164d63c433343a803147537ea7cd | subsection | 20 | 80 | Proof of | This Grassmannian bundle structure together with the motivic property of the E-polynomial imply the following equalityE\left( H^{[n,n+r]};t\right) =\sum _{m=1}^{\mu _{n}^{\max }} E\left( H^{[n]}_m;t\right)E\left( Gr_r(m);t\right)=\sum _{m=1}^{\mu _{n}^{\max }} E\left( H^{[n]}_m;t\right) {m r}_{t},where \mu _{n}^{\max }... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.018984463065862656,
-0.018145117908716202,
-0.00034622993553057313,
-0.01747364178299904,
-0.037877362221479416,
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0.024386795237660408,
-0.0011989284539595246,... | |
8ccea4d618900871661cd0c817b94159c1097806 | subsection | 21 | 80 | Proof of | We will consider all H_m^{[k]} and B_m^{[k]}, m,k\in \mathbb {N}, at the same time and we define following infinite matrices:\mathcal {X}:=\begin{pmatrix}E\left( H^{[j]}_i; t\right)\end{pmatrix}_{i\ge 1,j\ge 0}, \mathcal {B}:=\begin{pmatrix}E\left( B^{[j]}_i; t\right)\end{pmatrix}_{i\ge 1, j\ge 0},
\mathcal {R}:=\begin... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.016540581360459328,
0.03863538056612015,
-0.008773832581937313,
0.005569476168602705,
-0.0032005414832383394,
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-0.019897526130080223,
0.040466442704200745,
... | |
595f9c5ae32e32063c2be638bacfff3f92b12436 | subsection | 22 | 80 | Proof of | Then the matrices \mathcal {X} and \mathcal {B} may be expressed as products of matrices{\left\lbrace \begin{array}{ll}
\mathcal {X}&=\mathcal {G}^{-1}\mathcal {R}\\
\mathcal {B}&=\mathcal {X}\mathcal {A}^{-1}=\mathcal {G}^{-1}\mathcal {R}\mathcal {A}^{-1}
\end{array}\right.}Thus to compute the E-polynomials E\left( H^... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.017487453296780586,
0.017960499972105026,
-0.014076942577958107,
0.012573876418173313,
0.00840801652520895,
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0.03268597275018692,
0.054995447397232056,
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0.04452739283442497,
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0.01780790463089943,
0.0028096663299947977,
-0.... | |
0b5cf5b7ba7d17ed30ca52455095aecc1a0e53a1 | subsection | 23 | 80 | Proof of | To this end, we first need the generating function of E-polynomials of
the Hilbert scheme of points on the punctured plane Y_0.Proposition 14 (, , )
The E-polynomial E \left(Y_0^{[n]}; t \right) of the Hilbert scheme of points on the punctured complex plane Y_0 has the generating function\sum _{n=0}^\infty E \left(Y_0... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0073491",
"end": 404,
"openalex_id": "https://openalex.org/W579869728",
"raw": "L. Göttsche, Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.",... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.014195260591804981,
0.004973299335688353,
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0.024591896682977676,
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0.013615550473332405,
-0.008840573020279408,
0... | |
a8a98d7c70f625c07eec6623a28321266ce3555b | subsection | 24 | 80 | Proof of | \end{equation}
After the change of variable k=n-s, the right-hand side of the equation ((\ref {EpolyHnmotivic})) has the form of a doubly infinite summation
\begin{align}
\sum _{n=0}^{\infty } \left( \sum _{s=0}^n E\left( Y_0^{[s]}; t\right) E\left( B^{[n-s]};t \right) \right)q^n &\overset{\left(k=n-s \right)}{\scalebo... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.035310812294483185,
-0.0026170285418629646,
-0.012047487311065197,
-0.037935469299554825,
-0.02981734089553356,
-0.0651281550526619,
0.022218039259314537,
0.02810826152563095,
0.027467355132102966,
0.030442986637353897,
-0.05939052626490593,
0.03720300644636154,
0.03717248886823654,
-0.... | |
a4b820c40c97c2a0b2815bd2db70a3db6bb3cdc1 | subsection | 25 | 80 | Proof of | Recall that P\left(H^{[n]}; \sqrt{t}\right) has the generating function
\begin{equation*}
\sum _{n=0}^\infty P\left(H^{[n]}; \sqrt{t}\right)q^n=\prod _{d=1}^{\infty } \frac{1}{1-t^{d-1}q^d}.
\end{equation*} Thus we have
\sum _{n=0}^\infty E\left(H^{[n]};t \right) q^n=\prod _{d=1}^{\infty } \frac{1}{1-t^{d+1}q^d}=\left(... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.060639213770627975,
0.019531503319740295,
-0.007919413968920708,
-0.0190126970410347,
-0.0030117423739284277,
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0.0014352983562275767,
0.007156464736908674,
-0.025513025000691414,
0.022918997332453728,
-0.047669075429439545,
0.009903082624077797,
0.006881802808493376,
... | |
04562a473a6b3629f42160208ba8f8d1a7f5fd51 | subsection | 26 | 80 | Proof of | \end{align*}
}\begin{} We list some E-polynomials
E \left(Y_0^{[n]}; t \right) for 0\le n\le 8:
\begin{array}{|c|c|}\hline n& E \left(Y_0^{[n]}; t \right)\\\hline 0 & 1 \\\hline 1 & t^2-1 \\\hline 2 & t^4+t^3-t^2-t \\\hline 3 & t^6+t^5-2 t^3-t^2+t \\\hline 4 & t^8+t^7+t^6-t^5-3 t^4+t^2 \\\hline 5 & t^{10}+t^9+t^8-3 t^6... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.030723320320248604,
-0.009374121204018593,
-0.010678413324058056,
-0.004042542073875666,
0.024194233119487762,
-0.040181342512369156,
0.017909223213791847,
0.037343934178352356,
-0.0004240378621034324,
0.019663535058498383,
-0.032248806208372116,
0.055192138999700546,
-0.00308339181356132... | |
d064bd75b68d32997d7f0bfce41c808ff5d37893 | subsection | 27 | 80 | Proof of | Then from the definition of the generating function of {, we have
\begin{align*}
\sum _{n=0}^{\infty } { \left(Y_0^{[n]}; t \right)q^n &=
\sum _{n=0}^{\infty } E \left(Y_0^{[n]}; t^{-1} \right) t^{2n}q^n
\overset{\left(\tilde{ q}=t^2q\right)}{\scalebox {4.5}[1]{=}}
\prod _{d=1}^{\infty } \frac{1-t^{1 -d} \tilde{q}^d }{... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04289594665169716,
0.04735029488801956,
-0.0011898591183125973,
-0.004236966371536255,
0.013233369216322899,
-0.010136684402823448,
0.00021177681628614664,
0.05482504889369011,
-0.008153586648404598,
0.04387224465608597,
-0.026512503623962402,
0.010289231315255165,
-0.002211917657405138,
... | |
aa11fa9acc83a956e54eaa501bed762c5ddd956b | subsection | 28 | 80 | Proof of | The (i,j)-th entry of the matrix \mathcal {A}\mathcal {C} is given by the sum
\begin{align*}\left(\mathcal {A}\mathcal {C}\right)_{ij}
&=\sum _{k=i}^{j} E \left(Y_0^{[k-i]}; t \right)
{ \left(Y_0^{[j-k]}; t \right)\\
&=\sum _{l=0}^{j-i} E \left(Y_0^{[l]}; t \right)
{ \left(Y_0^{[j-i-l]}; t \right).
}
}Note that Proposi... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03246964141726494,
0.022444944828748703,
-0.016906186938285828,
-0.02671726606786251,
-0.0017041603568941355,
-0.018798213452100754,
-0.061887625604867935,
0.03111165389418602,
-0.0053022559732198715,
0.017272384837269783,
-0.019866295158863068,
0.01765384152531624,
0.006633542012423277,
... | |
b77c97e572afdeff6243b144f83606c27c222553 | subsection | 29 | 80 | Proof of | \begin{}[Proof of Theorem \ref {genEBmn}]
The E-polynomial E\left( B_m^{[n]};t\right) is equal to the (m,n)-th entry of the matrix product
\mathcal {B}=\mathcal {X}\mathcal {A}^{-1}=\mathcal {G}^{-1}\mathcal {R}\mathcal {A}^{-1} which is given by the sum
\begin{align*}
\sum _{k}\sum _{j}\mathcal {G}^{-1}_{mk}\mathcal {... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.024067994207143784,
-0.008622965775430202,
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0.010774890892207623,
-0.009569202549755573,
0.016986478120088577,
-0.028097132220864296,
0.03308776766061783,
-0.014010410755872726,... | |
bd474e2fb3159a50cb8728b8a237ec8b2a101d1e | subsection | 30 | 80 | Proof of | Then the generating function
\begin{align*}
\sum _{n=0}^{\infty }E\left( B_m^{[n]};t\right) q^n
&=
\sum _{n=0}^{\infty } \left( \sum _{k=0}^{\infty } \sum _{j=0}^{\infty }
(-1)^{k-m} t^{{k-m2}}
{k m}_{t}
E\left(H^{[j,j+k]}; t\right) {\left(Y_0^{[n-j]} ;t \right) q^n}
is equal to the product
\begin{align}
\prod _{d=1}^{... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03228325769305229,
0.026897627860307693,
-0.02810291014611721,
-0.0198337584733963,
-0.009756683371961117,
-0.017880896106362343,
-0.017209600657224655,
0.023113956674933434,
-0.04491583630442619,
0.02521938644349575,
-0.04918772354722023,
0.018231801688671112,
-0.013639523647725582,
0.... | |
0f30d707218fd576ab50ef23f91d1e14fa6c26dd | subsection | 31 | 80 | Proof of | \end{equation*}
Applying the Gauss^{\prime }s binomial formula with a=-t^{-k}, q=t, we obtain
\begin{align*}
\prod _{i=0}^{m-1}\left(1-t^{k-i}\right)&=
(-1)^mt^{mk-\binom{m}{2}}
\prod _{i=0}^{m-1}\left(1+(-t^{-k})t^{i}\right)
\\&=(-1)^mt^{mk-\binom{m}{2}}\sum _{i=0}^{m}t^{\binom{i}{2}}{m i}_{t}(-t^{-k})^i
\\&=\sum _{i=... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.020276261493563652,
0.019986383616924286,
-0.03481597453355789,
-0.03448032587766647,
-0.0147304218262434,
-0.0019032834097743034,
0.000698951305821538,
0.01021441537886858,
-0.003989648073911667,
0.025814473628997803,
-0.04360388219356537,
0.01583653874695301,
-0.025082148611545563,
-0... | |
fd5f3043ed842f9845cb1686ba647bf0d7e7c9d4 | subsection | 32 | 80 | Proof of | We substitute it into the generating function
\begin{align}
\sum _{n=0}^{\infty }E\left( B_m^{[n]};t\right) q^n
&=
\prod _{d=1}^{\infty } \frac{1}{1- t^{d-1}q^d } \cdot \sum _{k=0}^{\infty } \left((tq)^{\binom{k}{2}} (-1)^{k-m} t^{-km+\binom{m+1}{2}}
{k m}_{t}
\frac{1}{(tq)_k}\right)\\
\begin{split}
=&\prod _{d=1}^{\in... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04660444334149361,
0.03992050513625145,
-0.014222443103790283,
-0.0029413902666419744,
-0.03793668746948242,
0.004024859517812729,
0.03525090217590332,
0.009255269542336464,
0.0002324785600649193,
0.011017815209925175,
-0.032107315957546234,
0.032290440052747726,
-0.004993877839297056,
... | |
f4a7f396cb94fd904c83e7fe3b3d9e89fc1c22c7 | subsection | 33 | 80 | Proof of | \end{align*}Finally, we arrive at the result:
\begin{align*}
\sum _{n=0}^{\infty }E\left( B^{[n]}_m; t \right)q^n
&= \prod _{d=1}^{\infty } \frac{1}{1- t^{d-1}q^d } \cdot \prod _{i=0}^{m-1} \frac{1}{1- t^{i+1} }\cdot \sum _{a=0}^{m}\left( (-1)^{a} t^{m+\binom{a}{2}} {ma}_t
\prod _{k=0}^{\infty } (1-t^{-a} (tq)^k)\right... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03357894718647003,
0.019170526415109634,
-0.016026316210627556,
-0.04710210859775543,
-0.009745527058839798,
-0.007177500519901514,
0.022207895293831825,
0.03528842329978943,
-0.015751579776406288,
0.025764212012290955,
-0.04249263182282448,
0.024237895384430885,
-0.007082105614244938,
... | |
11a06c443b4d9fb9ac69d61ade7d4b8b44691a40 | subsection | 34 | 80 | Proof of | \begin{}
The E-polynomial of the refined stratum H^{[n]}_m has the generating function
\begin{align}
\sum _{n=0}^{\infty }E\left(H^{[n]}_m ;t\right) q^n&=
\prod _{i=1}^{m-1} \frac{1}{1-t^{i+1}}\cdot \sum _{a=1}^m\left( (-1)^a t^{\binom{a}{2}+m} {ma}_{t} \prod _{k=0}^{\infty } \frac{1-q^k t^{k-a}}{1-q^k t^{k+1}}\right).... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019270997494459152,
-0.005538695491850376,
-0.009337966330349445,
-0.03488004952669144,
0.0017632658127695322,
-0.03704670071601868,
-0.002580528613179922,
0.03777908906340599,
-0.014701193198561668,
0.026457570493221283,
-0.041929297149181366,
0.035765018314123154,
-0.01704331301152706,
... | |
f2320f4266dac0388f77a5ecc55b36457dcff48c | subsection | 35 | 80 | Proof of | Then the generating function of the E-polynomial E \left( H^{[n]}_m ;t \right)
\begin{align*}
\sum _{n=0}^{\infty }E\left(H^{[n]}_m; t\right) q^n=&
\sum _{n=0}^{\infty }\left(\sum _{j=0}^{n} E \left(( Y_0)^{[n-j]} ;t \right) E \left( B_m^{[j]} ;t \right) \right) q^n\\
=&
\left( \sum _{n=0}^{\infty }E\left(Y_0^{[n]}; t\... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.029746709391474724,
0.020487092435359955,
-0.0240872073918581,
-0.028328020125627518,
-0.02742799185216427,
-0.023919405415654182,
-0.016551373526453972,
0.031851861625909805,
-0.030494190752506256,
0.012569891288876534,
-0.04326239228248596,
0.02709238789975643,
-0.016871724277734756,
... | |
acbbc157fe7d3ab122db84b7781fed02b97ea2cf | subsection | 36 | 80 | Proof of | \end{align*}
\end{}\end{align*}
\end{}\begin{}
We list some examples of E\left( H^{[n]}_m;t \right):
\begin{equation} \begin{array}{|c|c|c| c|}
\hline n & m=1 & m=2 & m=3 \\\hline 1 & t^2-1 & 1 & 0 \\ \cline {1-1}
2 & t^4+t^3-t^2-t & t^2+t & 0 \\\cline {1-1}
3 & t^{6}+t^{5}-2 t^3-t^2+t & t^4+2 t^3+t^2-t-1 & 1 \\\cline ... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.02228672243654728,
0.009305202402174473,
-0.02677152492105961,
-0.0034398739226162434,
-0.02254604734480381,
-0.017054453492164612,
-0.02414776384830475,
0.047349750995635986,
-0.0011145267635583878,
0.004942435305565596,
-0.039020832628011703,
0.019434144720435143,
0.018762949854135513,
... | |
74b1bcd5cfebfe6a72410aec8a852a40cf3fe4ff | subsection | 37 | 80 | Proof of | \end{equation}
\begin{}[h]
\begin{array}{|c|cccc|}\hline n & m=2& m=3&m=4&m=5\\\hline 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
2 & 2 & 0 & 0 & 0 \\
3 & 2 & 1 & 0 & 0 \\
4 & 3 & 2 & 0 & 0 \\
5 & 2 & 5 & 0 & 0 \\
6 & 4 & 6 & 1 & 0 \\
7 & 2 & 11 & 2 & 0 \\\hline \end{array}
\caption {Examples of \chi \left( B^{[n]}_m\rig... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.01756073720753193,
-0.011557132005691528,
-0.03576226532459259,
-0.03173443302512169,
0.01299891248345375,
-0.022458214312791824,
-0.003758928505703807,
0.044672317802906036,
0.006735938601195812,
0.010222149081528187,
-0.03118518367409706,
0.019132202491164207,
-0.016645321622490883,
-0... | |
b8ecf5dadca20fbe2835fa266b85e485c41b2429 | subsection | 38 | 80 | Proof of | Indeed, each strata B^{[1]}_2, B^{[3]}_3 and B^{[6]}_4 contains a single monomial ideal:
1{16pt}\begin{array}{ccc}
(:y,~:x) & (:y^2,~:<xy>,~~:<x^2>) & (:<y^3>,~:<y^2x>,~~:<x^2y>,~~~:<x^3>) \\
B^{[1]}_2=\lbrace \mathfrak {m}\rbrace & B^{[3]}_3=\lbrace \langle x^2,xy,y^2\rangle \rbrace & B^{[6]}_4=\lbrace \langle x^3,x^2... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019327862188220024,
0.005857852753251791,
-0.04140159487724304,
-0.028496012091636658,
0.00866474024951458,
-0.03221818804740906,
0.029121460393071175,
-0.007066797465085983,
0.016658268868923187,
0.0034418697468936443,
-0.033682651817798615,
-0.009595285169780254,
0.027641741558909416,
... | |
d8c576e56949c6a1e4b36e4aa221e41e40a310a4 | subsection | 39 | 80 | Proof of | Göttsche,
Hilbert schemes of points on surfaces, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 483–494, Higher Ed. Press, Beijing.L. Göttsche,
Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.05306427925825119,
0.04326920583844185,
-0.010626587085425854,
-0.008727075532078743,
0.032284077256917953,
0.04812097176909447,
0.010046816430985928,
-0.01685912348330021,
0.017637236043810844,
0.01377718336880207,
-0.046747829765081406,
0.014479011297225952,
0.017866093665361404,
0.02... | |
71356354a505d10c1ac63acae5b31f3139892ec6 | subsection | 40 | 80 | Proof of | Thus for a fixed IH[n] with (I)=m, the set of JH[n+r] such that (I,J)H[n,n+r] is parameterized by the Grassmannian of r-dimensional subspaces of I/mICm. Over each stratum H[n]m of H[n], the projection map H[n,n+r]H[n] has fibers Gr(r,Cm) at each ideal IH[n]m.
Since Grassmannians are projective and smooth, their
E-polyn... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.015926921740174294,
-0.03725435212254524,
-0.028634896501898766,
-0.021632542833685875,
-0.005808596964925528,
-0.04314304143190384,
-0.004977162927389145,
-0.013547036796808243,
-0.0017753788270056248,
0.02218174748122692,
-0.05488990619778633,
0.021388452500104904,
0.01263169664889574,
... | |
fa1d1e81d4587b490b2ab79d1be168ee12ce2e17 | subsection | 41 | 80 | Proof of | We will consider all H_m^{[k]} and B_m^{[k]}, m,k\in \mathbb {N}, at the same time and we define following infinite matrices:\mathcal {X}:=\begin{pmatrix}E\left( H^{[j]}_i; t\right)\end{pmatrix}_{i\ge 1,j\ge 0}, \mathcal {B}:=\begin{pmatrix}E\left( B^{[j]}_i; t\right)\end{pmatrix}_{i\ge 1, j\ge 0},
\mathcal {R}:=\begin... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.016540581360459328,
0.03863538056612015,
-0.008773832581937313,
0.005569476168602705,
-0.0032005414832383394,
-0.013763472437858582,
0.010215792804956436,
0.041168347001075745,
0.013198895379900932,
0.02877817116677761,
-0.03216563165187836,
-0.019897526130080223,
0.040466442704200745,
... | |
1650b588f4644f544473bd7735af91cde3fbe45c | subsection | 42 | 80 | Proof of | Then the matrices \mathcal {X} and \mathcal {B} may be expressed as products of matrices{\left\lbrace \begin{array}{ll}
\mathcal {X}&=\mathcal {G}^{-1}\mathcal {R}\\
\mathcal {B}&=\mathcal {X}\mathcal {A}^{-1}=\mathcal {G}^{-1}\mathcal {R}\mathcal {A}^{-1}
\end{array}\right.}Thus to compute the E-polynomials E\left( H^... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.017487453296780586,
0.017960499972105026,
-0.014076942577958107,
0.012573876418173313,
0.00840801652520895,
-0.033296357840299606,
0.03268597275018692,
0.054995447397232056,
-0.011452298611402512,
0.04452739283442497,
-0.03604307770729065,
0.01780790463089943,
0.0028096663299947977,
-0.... | |
ad7a5b1e1a100618ca282f150d6e1dc58333e5b1 | subsection | 43 | 80 | Proof of | To this end, we first need the generating function of E-polynomials of
the Hilbert scheme of points on the punctured plane Y_0.Proposition 14 (, , )
The E-polynomial E \left(Y_0^{[n]}; t \right) of the Hilbert scheme of points on the punctured complex plane Y_0 has the generating function\sum _{n=0}^\infty E \left(Y_0... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0073491",
"end": 404,
"openalex_id": "https://openalex.org/W579869728",
"raw": "L. Göttsche, Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.",... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.014195260591804981,
0.004973299335688353,
-0.019725386053323746,
-0.005797097459435463,
0.00887871254235506,
-0.03356214240193367,
0.004557586275041103,
0.03325703367590904,
-0.018382901325821877,
0.024591896682977676,
-0.05412658303976059,
0.013615550473332405,
-0.008840573020279408,
0... | |
795acbcab00f4935f824f85bc878c68eb94cdcc1 | subsection | 44 | 80 | Proof of | \end{equation}
After the change of variable k=n-s, the right-hand side of the equation ((\ref {EpolyHnmotivic})) has the form of a doubly infinite summation
\begin{align}
\sum _{n=0}^{\infty } \left( \sum _{s=0}^n E\left( Y_0^{[s]}; t\right) E\left( B^{[n-s]};t \right) \right)q^n &\overset{\left(k=n-s \right)}{\scalebo... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.035310812294483185,
-0.0026170285418629646,
-0.012047487311065197,
-0.037935469299554825,
-0.02981734089553356,
-0.0651281550526619,
0.022218039259314537,
0.02810826152563095,
0.027467355132102966,
0.030442986637353897,
-0.05939052626490593,
0.03720300644636154,
0.03717248886823654,
-0.... | |
398754a20d1010cfbe8e2943f1acdb9a607cccd5 | subsection | 45 | 80 | Proof of | Recall that P\left(H^{[n]}; \sqrt{t}\right) has the generating function
\begin{equation*}
\sum _{n=0}^\infty P\left(H^{[n]}; \sqrt{t}\right)q^n=\prod _{d=1}^{\infty } \frac{1}{1-t^{d-1}q^d}.
\end{equation*} Thus we have
\sum _{n=0}^\infty E\left(H^{[n]};t \right) q^n=\prod _{d=1}^{\infty } \frac{1}{1-t^{d+1}q^d}=\left(... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.060639213770627975,
0.019531503319740295,
-0.007919413968920708,
-0.0190126970410347,
-0.0030117423739284277,
-0.00842296052724123,
0.0014352983562275767,
0.007156464736908674,
-0.025513025000691414,
0.022918997332453728,
-0.047669075429439545,
0.009903082624077797,
0.006881802808493376,
... | |
ddc99590222dc21bcdcad8c44bc894f70c097c4b | subsection | 46 | 80 | Proof of | \end{align*}
}\begin{} We list some E-polynomials
E \left(Y_0^{[n]}; t \right) for 0\le n\le 8:
\begin{array}{|c|c|}\hline n& E \left(Y_0^{[n]}; t \right)\\\hline 0 & 1 \\\hline 1 & t^2-1 \\\hline 2 & t^4+t^3-t^2-t \\\hline 3 & t^6+t^5-2 t^3-t^2+t \\\hline 4 & t^8+t^7+t^6-t^5-3 t^4+t^2 \\\hline 5 & t^{10}+t^9+t^8-3 t^6... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.030723320320248604,
-0.009374121204018593,
-0.010678413324058056,
-0.004042542073875666,
0.024194233119487762,
-0.040181342512369156,
0.017909223213791847,
0.037343934178352356,
-0.0004240378621034324,
0.019663535058498383,
-0.032248806208372116,
0.055192138999700546,
-0.00308339181356132... | |
d9f36fb55a93d4054fdf5d9d14892b06eb232b7c | subsection | 47 | 80 | Proof of | Then from the definition of the generating function of {, we have
\begin{align*}
\sum _{n=0}^{\infty } { \left(Y_0^{[n]}; t \right)q^n &=
\sum _{n=0}^{\infty } E \left(Y_0^{[n]}; t^{-1} \right) t^{2n}q^n
\overset{\left(\tilde{ q}=t^2q\right)}{\scalebox {4.5}[1]{=}}
\prod _{d=1}^{\infty } \frac{1-t^{1 -d} \tilde{q}^d }{... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04289594665169716,
0.04735029488801956,
-0.0011898591183125973,
-0.004236966371536255,
0.013233369216322899,
-0.010136684402823448,
0.00021177681628614664,
0.05482504889369011,
-0.008153586648404598,
0.04387224465608597,
-0.026512503623962402,
0.010289231315255165,
-0.002211917657405138,
... | |
25437e2f3aa2028c68edd05ec603107bd4825e8b | subsection | 48 | 80 | Proof of | The (i,j)-th entry of the matrix \mathcal {A}\mathcal {C} is given by the sum
\begin{align*}\left(\mathcal {A}\mathcal {C}\right)_{ij}
&=\sum _{k=i}^{j} E \left(Y_0^{[k-i]}; t \right)
{ \left(Y_0^{[j-k]}; t \right)\\
&=\sum _{l=0}^{j-i} E \left(Y_0^{[l]}; t \right)
{ \left(Y_0^{[j-i-l]}; t \right).
}
}Note that Proposi... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03246964141726494,
0.022444944828748703,
-0.016906186938285828,
-0.02671726606786251,
-0.0017041603568941355,
-0.018798213452100754,
-0.061887625604867935,
0.03111165389418602,
-0.0053022559732198715,
0.017272384837269783,
-0.019866295158863068,
0.01765384152531624,
0.006633542012423277,
... | |
890bed577d9b0d06cd5391cf9d5caff017f8c08b | subsection | 49 | 80 | Proof of | \begin{}[Proof of Theorem \ref {genEBmn}]
The E-polynomial E\left( B_m^{[n]};t\right) is equal to the (m,n)-th entry of the matrix product
\mathcal {B}=\mathcal {X}\mathcal {A}^{-1}=\mathcal {G}^{-1}\mathcal {R}\mathcal {A}^{-1} which is given by the sum
\begin{align*}
\sum _{k}\sum _{j}\mathcal {G}^{-1}_{mk}\mathcal {... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.024067994207143784,
-0.008622965775430202,
-0.02602151408791542,
-0.012011104263365269,
-0.016147075220942497,
-0.03711690753698349,
-0.010416236706078053,
0.010774890892207623,
-0.009569202549755573,
0.016986478120088577,
-0.028097132220864296,
0.03308776766061783,
-0.014010410755872726,... | |
c3c074fcb3087ce3d0dff1ec095fe757334bdee6 | subsection | 50 | 80 | Proof of | Then the generating function
\begin{align*}
\sum _{n=0}^{\infty }E\left( B_m^{[n]};t\right) q^n
&=
\sum _{n=0}^{\infty } \left( \sum _{k=0}^{\infty } \sum _{j=0}^{\infty }
(-1)^{k-m} t^{{k-m2}}
{k m}_{t}
E\left(H^{[j,j+k]}; t\right) {\left(Y_0^{[n-j]} ;t \right) q^n}
is equal to the product
\begin{align}
\prod _{d=1}^{... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03228325769305229,
0.026897627860307693,
-0.02810291014611721,
-0.0198337584733963,
-0.009756683371961117,
-0.017880896106362343,
-0.017209600657224655,
0.023113956674933434,
-0.04491583630442619,
0.02521938644349575,
-0.04918772354722023,
0.018231801688671112,
-0.013639523647725582,
0.... | |
835cbd774d6495258ff301e4ececc71ba389eed0 | subsection | 51 | 80 | Proof of | \end{equation*}
Applying the Gauss^{\prime }s binomial formula with a=-t^{-k}, q=t, we obtain
\begin{align*}
\prod _{i=0}^{m-1}\left(1-t^{k-i}\right)&=
(-1)^mt^{mk-\binom{m}{2}}
\prod _{i=0}^{m-1}\left(1+(-t^{-k})t^{i}\right)
\\&=(-1)^mt^{mk-\binom{m}{2}}\sum _{i=0}^{m}t^{\binom{i}{2}}{m i}_{t}(-t^{-k})^i
\\&=\sum _{i=... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.020276261493563652,
0.019986383616924286,
-0.03481597453355789,
-0.03448032587766647,
-0.0147304218262434,
-0.0019032834097743034,
0.000698951305821538,
0.01021441537886858,
-0.003989648073911667,
0.025814473628997803,
-0.04360388219356537,
0.01583653874695301,
-0.025082148611545563,
-0... | |
4cd22ace36e28f39f28cd99e78d0f8ac93c20d19 | subsection | 52 | 80 | Proof of | We substitute it into the generating function
\begin{align}
\sum _{n=0}^{\infty }E\left( B_m^{[n]};t\right) q^n
&=
\prod _{d=1}^{\infty } \frac{1}{1- t^{d-1}q^d } \cdot \sum _{k=0}^{\infty } \left((tq)^{\binom{k}{2}} (-1)^{k-m} t^{-km+\binom{m+1}{2}}
{k m}_{t}
\frac{1}{(tq)_k}\right)\\
\begin{split}
=&\prod _{d=1}^{\in... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04660444334149361,
0.03992050513625145,
-0.014222443103790283,
-0.0029413902666419744,
-0.03793668746948242,
0.004024859517812729,
0.03525090217590332,
0.009255269542336464,
0.0002324785600649193,
0.011017815209925175,
-0.032107315957546234,
0.032290440052747726,
-0.004993877839297056,
... | |
70714bc7fc551cc35390ce823fc4e756c9241a01 | subsection | 53 | 80 | Proof of | \end{align*}Finally, we arrive at the result:
\begin{align*}
\sum _{n=0}^{\infty }E\left( B^{[n]}_m; t \right)q^n
&= \prod _{d=1}^{\infty } \frac{1}{1- t^{d-1}q^d } \cdot \prod _{i=0}^{m-1} \frac{1}{1- t^{i+1} }\cdot \sum _{a=0}^{m}\left( (-1)^{a} t^{m+\binom{a}{2}} {ma}_t
\prod _{k=0}^{\infty } (1-t^{-a} (tq)^k)\right... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03357894718647003,
0.019170526415109634,
-0.016026316210627556,
-0.04710210859775543,
-0.009745527058839798,
-0.007177500519901514,
0.022207895293831825,
0.03528842329978943,
-0.015751579776406288,
0.025764212012290955,
-0.04249263182282448,
0.024237895384430885,
-0.007082105614244938,
... | |
b615a124a9381ca91bbc657f182fe15d482e2d46 | subsection | 54 | 80 | Proof of | \begin{}
The E-polynomial of the refined stratum H^{[n]}_m has the generating function
\begin{align}
\sum _{n=0}^{\infty }E\left(H^{[n]}_m ;t\right) q^n&=
\prod _{i=1}^{m-1} \frac{1}{1-t^{i+1}}\cdot \sum _{a=1}^m\left( (-1)^a t^{\binom{a}{2}+m} {ma}_{t} \prod _{k=0}^{\infty } \frac{1-q^k t^{k-a}}{1-q^k t^{k+1}}\right).... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019270997494459152,
-0.005538695491850376,
-0.009337966330349445,
-0.03488004952669144,
0.0017632658127695322,
-0.03704670071601868,
-0.002580528613179922,
0.03777908906340599,
-0.014701193198561668,
0.026457570493221283,
-0.041929297149181366,
0.035765018314123154,
-0.01704331301152706,
... | |
2a6f6cc22ae07b2485f2ab126682f09554b60fff | subsection | 55 | 80 | Proof of | Then the generating function of the E-polynomial E \left( H^{[n]}_m ;t \right)
\begin{align*}
\sum _{n=0}^{\infty }E\left(H^{[n]}_m; t\right) q^n=&
\sum _{n=0}^{\infty }\left(\sum _{j=0}^{n} E \left(( Y_0)^{[n-j]} ;t \right) E \left( B_m^{[j]} ;t \right) \right) q^n\\
=&
\left( \sum _{n=0}^{\infty }E\left(Y_0^{[n]}; t\... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.029746709391474724,
0.020487092435359955,
-0.0240872073918581,
-0.028328020125627518,
-0.02742799185216427,
-0.023919405415654182,
-0.016551373526453972,
0.031851861625909805,
-0.030494190752506256,
0.012569891288876534,
-0.04326239228248596,
0.02709238789975643,
-0.016871724277734756,
... | |
c591bb8074b6b067d149ac4666d465be7ae949d7 | subsection | 56 | 80 | Proof of | \end{align*}
\end{}\end{align*}
\end{}\begin{}
We list some examples of E\left( H^{[n]}_m;t \right):
\begin{equation} \begin{array}{|c|c|c| c|}
\hline n & m=1 & m=2 & m=3 \\\hline 1 & t^2-1 & 1 & 0 \\ \cline {1-1}
2 & t^4+t^3-t^2-t & t^2+t & 0 \\\cline {1-1}
3 & t^{6}+t^{5}-2 t^3-t^2+t & t^4+2 t^3+t^2-t-1 & 1 \\\cline ... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.02228672243654728,
0.009305202402174473,
-0.02677152492105961,
-0.0034398739226162434,
-0.02254604734480381,
-0.017054453492164612,
-0.02414776384830475,
0.047349750995635986,
-0.0011145267635583878,
0.004942435305565596,
-0.039020832628011703,
0.019434144720435143,
0.018762949854135513,
... | |
d2aed0fdf448048290a6eff51efea11a1f861a06 | subsection | 57 | 80 | Proof of | \end{equation}
\begin{}[h]
\begin{array}{|c|cccc|}\hline n & m=2& m=3&m=4&m=5\\\hline 0 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
2 & 2 & 0 & 0 & 0 \\
3 & 2 & 1 & 0 & 0 \\
4 & 3 & 2 & 0 & 0 \\
5 & 2 & 5 & 0 & 0 \\
6 & 4 & 6 & 1 & 0 \\
7 & 2 & 11 & 2 & 0 \\\hline \end{array}
\caption {Examples of \chi \left( B^{[n]}_m\rig... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
0.01756073720753193,
-0.011557132005691528,
-0.03576226532459259,
-0.03173443302512169,
0.01299891248345375,
-0.022458214312791824,
-0.003758928505703807,
0.044672317802906036,
0.006735938601195812,
0.010222149081528187,
-0.03118518367409706,
0.019132202491164207,
-0.016645321622490883,
-0... | |
9aa10c9237abb2dfe59f66215dca0a92f3d7fc0c | subsection | 58 | 80 | Proof of | Indeed, each strata B^{[1]}_2, B^{[3]}_3 and B^{[6]}_4 contains a single monomial ideal:
1{16pt}\begin{array}{ccc}
(:y,~:x) & (:y^2,~:<xy>,~~:<x^2>) & (:<y^3>,~:<y^2x>,~~:<x^2y>,~~~:<x^3>) \\
B^{[1]}_2=\lbrace \mathfrak {m}\rbrace & B^{[3]}_3=\lbrace \langle x^2,xy,y^2\rangle \rbrace & B^{[6]}_4=\lbrace \langle x^3,x^2... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019327862188220024,
0.005857852753251791,
-0.04140159487724304,
-0.028496012091636658,
0.00866474024951458,
-0.03221818804740906,
0.029121460393071175,
-0.007066797465085983,
0.016658268868923187,
0.0034418697468936443,
-0.033682651817798615,
-0.009595285169780254,
0.027641741558909416,
... | |
4c3f08a07416173b0a120cf793f4f56f5e10c5f3 | subsection | 59 | 80 | Proof of | Göttsche,
Hilbert schemes of points on surfaces, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), 483–494, Higher Ed. Press, Beijing.L. Göttsche,
Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.05306427925825119,
0.04326920583844185,
-0.010626587085425854,
-0.008727075532078743,
0.032284077256917953,
0.04812097176909447,
0.010046816430985928,
-0.01685912348330021,
0.017637236043810844,
0.01377718336880207,
-0.046747829765081406,
0.014479011297225952,
0.017866093665361404,
0.02... | |
a3e3a8f869517a3c990fa4fb70b54445cdc5f955 | subsection | 60 | 80 | Proof of | Thus for a fixed IH[n] with (I)=m, the set of JH[n+r] such that (I,J)H[n,n+r] is parameterized by the Grassmannian of r-dimensional subspaces of I/mICm. Over each stratum H[n]m of H[n], the projection map H[n,n+r]H[n] has fibers Gr(r,Cm) at each ideal IH[n]m.
Since Grassmannians are projective and smooth, their
E-polyn... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.015926921740174294,
-0.03725435212254524,
-0.028634896501898766,
-0.021632542833685875,
-0.005808596964925528,
-0.04314304143190384,
-0.004977162927389145,
-0.013547036796808243,
-0.0017753788270056248,
0.02218174748122692,
-0.05488990619778633,
0.021388452500104904,
0.01263169664889574,
... | |
1fa2515fca0ab3c22a4cca034bfa29a3c525db2d | subsection | 61 | 80 | Proof of | We will consider all H_m^{[k]} and B_m^{[k]}, m,k\in \mathbb {N}, at the same time and we define following infinite matrices:\mathcal {X}:=\begin{pmatrix}E\left( H^{[j]}_i; t\right)\end{pmatrix}_{i\ge 1,j\ge 0}, \mathcal {B}:=\begin{pmatrix}E\left( B^{[j]}_i; t\right)\end{pmatrix}_{i\ge 1, j\ge 0},
\mathcal {R}:=\begin... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.016540581360459328,
0.03863538056612015,
-0.008773832581937313,
0.005569476168602705,
-0.0032005414832383394,
-0.013763472437858582,
0.010215792804956436,
0.041168347001075745,
0.013198895379900932,
0.02877817116677761,
-0.03216563165187836,
-0.019897526130080223,
0.040466442704200745,
... | |
2accf62f7cc5ddfdf5b48c973170f00020503cfe | subsection | 62 | 80 | Proof of | Then the matrices \mathcal {X} and \mathcal {B} may be expressed as products of matrices{\left\lbrace \begin{array}{ll}
\mathcal {X}&=\mathcal {G}^{-1}\mathcal {R}\\
\mathcal {B}&=\mathcal {X}\mathcal {A}^{-1}=\mathcal {G}^{-1}\mathcal {R}\mathcal {A}^{-1}
\end{array}\right.}Thus to compute the E-polynomials E\left( H^... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.017487453296780586,
0.017960499972105026,
-0.014076942577958107,
0.012573876418173313,
0.00840801652520895,
-0.033296357840299606,
0.03268597275018692,
0.054995447397232056,
-0.011452298611402512,
0.04452739283442497,
-0.03604307770729065,
0.01780790463089943,
0.0028096663299947977,
-0.... | |
58da4087dcb53be0103522d41276a702569b908d | subsection | 63 | 80 | Proof of | To this end, we first need the generating function of E-polynomials of
the Hilbert scheme of points on the punctured plane Y_0.Proposition 14 (, , )
The E-polynomial E \left(Y_0^{[n]}; t \right) of the Hilbert scheme of points on the punctured complex plane Y_0 has the generating function\sum _{n=0}^\infty E \left(Y_0... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0073491",
"end": 404,
"openalex_id": "https://openalex.org/W579869728",
"raw": "L. Göttsche, Hilbert schemes of zero-dimensional subschemes of smooth varieties, Lecture Notes in Mathematics, 1572, Springer-Verlag, Berlin, 1994.",... | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.014195260591804981,
0.004973299335688353,
-0.019725386053323746,
-0.005797097459435463,
0.00887871254235506,
-0.03356214240193367,
0.004557586275041103,
0.03325703367590904,
-0.018382901325821877,
0.024591896682977676,
-0.05412658303976059,
0.013615550473332405,
-0.008840573020279408,
0... | |
d1d212ab40f0f9fad4b914a8d9ccd48fd682ba7d | subsection | 64 | 80 | Proof of | \end{equation}
After the change of variable k=n-s, the right-hand side of the equation ((\ref {EpolyHnmotivic})) has the form of a doubly infinite summation
\begin{align}
\sum _{n=0}^{\infty } \left( \sum _{s=0}^n E\left( Y_0^{[s]}; t\right) E\left( B^{[n-s]};t \right) \right)q^n &\overset{\left(k=n-s \right)}{\scalebo... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.035310812294483185,
-0.0026170285418629646,
-0.012047487311065197,
-0.037935469299554825,
-0.02981734089553356,
-0.0651281550526619,
0.022218039259314537,
0.02810826152563095,
0.027467355132102966,
0.030442986637353897,
-0.05939052626490593,
0.03720300644636154,
0.03717248886823654,
-0.... | |
0ef2427fb32973d2bdab391fa5484c2608ed8f3c | subsection | 65 | 80 | Proof of | Recall that P\left(H^{[n]}; \sqrt{t}\right) has the generating function
\begin{equation*}
\sum _{n=0}^\infty P\left(H^{[n]}; \sqrt{t}\right)q^n=\prod _{d=1}^{\infty } \frac{1}{1-t^{d-1}q^d}.
\end{equation*} Thus we have
\sum _{n=0}^\infty E\left(H^{[n]};t \right) q^n=\prod _{d=1}^{\infty } \frac{1}{1-t^{d+1}q^d}=\left(... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.060639213770627975,
0.019531503319740295,
-0.007919413968920708,
-0.0190126970410347,
-0.0030117423739284277,
-0.00842296052724123,
0.0014352983562275767,
0.007156464736908674,
-0.025513025000691414,
0.022918997332453728,
-0.047669075429439545,
0.009903082624077797,
0.006881802808493376,
... | |
a9f162ec8886a5ee118235f758a1eba7a5353ca0 | subsection | 66 | 80 | Proof of | \end{align*}
}\begin{} We list some E-polynomials
E \left(Y_0^{[n]}; t \right) for 0\le n\le 8:
\begin{array}{|c|c|}\hline n& E \left(Y_0^{[n]}; t \right)\\\hline 0 & 1 \\\hline 1 & t^2-1 \\\hline 2 & t^4+t^3-t^2-t \\\hline 3 & t^6+t^5-2 t^3-t^2+t \\\hline 4 & t^8+t^7+t^6-t^5-3 t^4+t^2 \\\hline 5 & t^{10}+t^9+t^8-3 t^6... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.030723320320248604,
-0.009374121204018593,
-0.010678413324058056,
-0.004042542073875666,
0.024194233119487762,
-0.040181342512369156,
0.017909223213791847,
0.037343934178352356,
-0.0004240378621034324,
0.019663535058498383,
-0.032248806208372116,
0.055192138999700546,
-0.00308339181356132... | |
a331ae2cd59331f3f3e3881912968c4b73d6ad76 | subsection | 67 | 80 | Proof of | Then from the definition of the generating function of {, we have
\begin{align*}
\sum _{n=0}^{\infty } { \left(Y_0^{[n]}; t \right)q^n &=
\sum _{n=0}^{\infty } E \left(Y_0^{[n]}; t^{-1} \right) t^{2n}q^n
\overset{\left(\tilde{ q}=t^2q\right)}{\scalebox {4.5}[1]{=}}
\prod _{d=1}^{\infty } \frac{1-t^{1 -d} \tilde{q}^d }{... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04289594665169716,
0.04735029488801956,
-0.0011898591183125973,
-0.004236966371536255,
0.013233369216322899,
-0.010136684402823448,
0.00021177681628614664,
0.05482504889369011,
-0.008153586648404598,
0.04387224465608597,
-0.026512503623962402,
0.010289231315255165,
-0.002211917657405138,
... | |
d50c5372a15b176d730089777781fd54f7d0d39f | subsection | 68 | 80 | Proof of | The (i,j)-th entry of the matrix \mathcal {A}\mathcal {C} is given by the sum
\begin{align*}\left(\mathcal {A}\mathcal {C}\right)_{ij}
&=\sum _{k=i}^{j} E \left(Y_0^{[k-i]}; t \right)
{ \left(Y_0^{[j-k]}; t \right)\\
&=\sum _{l=0}^{j-i} E \left(Y_0^{[l]}; t \right)
{ \left(Y_0^{[j-i-l]}; t \right).
}
}Note that Proposi... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03246964141726494,
0.022444944828748703,
-0.016906186938285828,
-0.02671726606786251,
-0.0017041603568941355,
-0.018798213452100754,
-0.061887625604867935,
0.03111165389418602,
-0.0053022559732198715,
0.017272384837269783,
-0.019866295158863068,
0.01765384152531624,
0.006633542012423277,
... | |
be2edaa73d98b309180fb1cc073827f694afb396 | subsection | 69 | 80 | Proof of | \begin{}[Proof of Theorem \ref {genEBmn}]
The E-polynomial E\left( B_m^{[n]};t\right) is equal to the (m,n)-th entry of the matrix product
\mathcal {B}=\mathcal {X}\mathcal {A}^{-1}=\mathcal {G}^{-1}\mathcal {R}\mathcal {A}^{-1} which is given by the sum
\begin{align*}
\sum _{k}\sum _{j}\mathcal {G}^{-1}_{mk}\mathcal {... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.024067994207143784,
-0.008622965775430202,
-0.02602151408791542,
-0.012011104263365269,
-0.016147075220942497,
-0.03711690753698349,
-0.010416236706078053,
0.010774890892207623,
-0.009569202549755573,
0.016986478120088577,
-0.028097132220864296,
0.03308776766061783,
-0.014010410755872726,... | |
e8612d66221ad2eaaaf95155b8d924aa6172dacd | subsection | 70 | 80 | Proof of | Then the generating function
\begin{align*}
\sum _{n=0}^{\infty }E\left( B_m^{[n]};t\right) q^n
&=
\sum _{n=0}^{\infty } \left( \sum _{k=0}^{\infty } \sum _{j=0}^{\infty }
(-1)^{k-m} t^{{k-m2}}
{k m}_{t}
E\left(H^{[j,j+k]}; t\right) {\left(Y_0^{[n-j]} ;t \right) q^n}
is equal to the product
\begin{align}
\prod _{d=1}^{... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03228325769305229,
0.026897627860307693,
-0.02810291014611721,
-0.0198337584733963,
-0.009756683371961117,
-0.017880896106362343,
-0.017209600657224655,
0.023113956674933434,
-0.04491583630442619,
0.02521938644349575,
-0.04918772354722023,
0.018231801688671112,
-0.013639523647725582,
0.... | |
0588c3779ac74f9a809127bc8c5be822a9fba482 | subsection | 71 | 80 | Proof of | \end{equation*}
Applying the Gauss^{\prime }s binomial formula with a=-t^{-k}, q=t, we obtain
\begin{align*}
\prod _{i=0}^{m-1}\left(1-t^{k-i}\right)&=
(-1)^mt^{mk-\binom{m}{2}}
\prod _{i=0}^{m-1}\left(1+(-t^{-k})t^{i}\right)
\\&=(-1)^mt^{mk-\binom{m}{2}}\sum _{i=0}^{m}t^{\binom{i}{2}}{m i}_{t}(-t^{-k})^i
\\&=\sum _{i=... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.020276261493563652,
0.019986383616924286,
-0.03481597453355789,
-0.03448032587766647,
-0.0147304218262434,
-0.0019032834097743034,
0.000698951305821538,
0.01021441537886858,
-0.003989648073911667,
0.025814473628997803,
-0.04360388219356537,
0.01583653874695301,
-0.025082148611545563,
-0... | |
ec2a6ce7ba328dbd48277585291d67ab4b301419 | subsection | 72 | 80 | Proof of | We substitute it into the generating function
\begin{align}
\sum _{n=0}^{\infty }E\left( B_m^{[n]};t\right) q^n
&=
\prod _{d=1}^{\infty } \frac{1}{1- t^{d-1}q^d } \cdot \sum _{k=0}^{\infty } \left((tq)^{\binom{k}{2}} (-1)^{k-m} t^{-km+\binom{m+1}{2}}
{k m}_{t}
\frac{1}{(tq)_k}\right)\\
\begin{split}
=&\prod _{d=1}^{\in... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.04660444334149361,
0.03992050513625145,
-0.014222443103790283,
-0.0029413902666419744,
-0.03793668746948242,
0.004024859517812729,
0.03525090217590332,
0.009255269542336464,
0.0002324785600649193,
0.011017815209925175,
-0.032107315957546234,
0.032290440052747726,
-0.004993877839297056,
... | |
6bc847421304acf4230470b3da4d2580ebbbc5a5 | subsection | 73 | 80 | Proof of | \end{align*}Finally, we arrive at the result:
\begin{align*}
\sum _{n=0}^{\infty }E\left( B^{[n]}_m; t \right)q^n
&= \prod _{d=1}^{\infty } \frac{1}{1- t^{d-1}q^d } \cdot \prod _{i=0}^{m-1} \frac{1}{1- t^{i+1} }\cdot \sum _{a=0}^{m}\left( (-1)^{a} t^{m+\binom{a}{2}} {ma}_t
\prod _{k=0}^{\infty } (1-t^{-a} (tq)^k)\right... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.03357894718647003,
0.019170526415109634,
-0.016026316210627556,
-0.04710210859775543,
-0.009745527058839798,
-0.007177500519901514,
0.022207895293831825,
0.03528842329978943,
-0.015751579776406288,
0.025764212012290955,
-0.04249263182282448,
0.024237895384430885,
-0.007082105614244938,
... | |
0d32d9112438cbc393842556dc9ff7fc1e4e07d1 | subsection | 74 | 80 | Proof of | \begin{}
The E-polynomial of the refined stratum H^{[n]}_m has the generating function
\begin{align}
\sum _{n=0}^{\infty }E\left(H^{[n]}_m ;t\right) q^n&=
\prod _{i=1}^{m-1} \frac{1}{1-t^{i+1}}\cdot \sum _{a=1}^m\left( (-1)^a t^{\binom{a}{2}+m} {ma}_{t} \prod _{k=0}^{\infty } \frac{1-q^k t^{k-a}}{1-q^k t^{k+1}}\right).... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.019270997494459152,
-0.005538695491850376,
-0.009337966330349445,
-0.03488004952669144,
0.0017632658127695322,
-0.03704670071601868,
-0.002580528613179922,
0.03777908906340599,
-0.014701193198561668,
0.026457570493221283,
-0.041929297149181366,
0.035765018314123154,
-0.01704331301152706,
... | |
004cad5b5d55b0b9a64c5b0dd3b2308b5f135ae1 | subsection | 75 | 80 | Proof of | Then the generating function of the E-polynomial E \left( H^{[n]}_m ;t \right)
\begin{align*}
\sum _{n=0}^{\infty }E\left(H^{[n]}_m; t\right) q^n=&
\sum _{n=0}^{\infty }\left(\sum _{j=0}^{n} E \left(( Y_0)^{[n-j]} ;t \right) E \left( B_m^{[j]} ;t \right) \right) q^n\\
=&
\left( \sum _{n=0}^{\infty }E\left(Y_0^{[n]}; t\... | {
"cite_spans": []
} | 1806.03955 | Refined Hilbert schemes, E-polynomials, and the number of generators of
finite colength ideals in the plane | [
"Yi-Ning Hsiao",
"Andras Szenes"
] | [
"math.AG"
] | 2,018 | en | Mathematics | [
-0.029746709391474724,
0.020487092435359955,
-0.0240872073918581,
-0.028328020125627518,
-0.02742799185216427,
-0.023919405415654182,
-0.016551373526453972,
0.031851861625909805,
-0.030494190752506256,
0.012569891288876534,
-0.04326239228248596,
0.02709238789975643,
-0.016871724277734756,
... |
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