文章基本信息

标题：A Generalized Sylvester-Gallai Type Theorem for Quadratic Polynomials
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作者：Shir Peleg ; Amir Shpilka
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2020
卷号：169
页码：8:1-8:33
DOI：10.4230/LIPIcs.CCC.2020.8
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Î£^{[3]}Î Î£Î ^{[2]} circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials ð'¬ satisfy that for every two polynomials Qâ,,Qâ,, â^^ ð'¬ there is a subset ð'¦ âS, ð'¬, such that Qâ,,Qâ,, â^ ð'¦ and whenever Qâ, and Qâ,, vanish then â^_{Q_iâ^^ð'¦} Q_i vanishes, then the linear span of the polynomials in ð'¬ has dimension O(1). This extends the earlier result [Amir Shpilka, 2019] that showed a similar conclusion when ð'¦ = 1. An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generated by the two quadratics. This step extends a result from [Amir Shpilka, 2019] that studied the case when one quadratic polynomial is in the radical of two other quadratics.
关键词：Algebraic computation; Computational complexity; Computational geometry