文章基本信息

标题：On the Polynomial Parity Argument Complexity of the Combinatorial Nullstellensatz
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作者：Aleksandrs Belovs ; G{\'a}bor Ivanyos ; Youming Qiao 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：79
页码：30:1-30:24
DOI：10.4230/LIPIcs.CCC.2017.30
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：The complexity class PPA consists of NP-search problems which are reducible to the parity principle in undirected graphs. It contains a wide variety of interesting problems from graph theory, combinatorics, algebra and number theory, but only a few of these are known to be complete in the class. Before this work, the known complete problems were all discretizations or combinatorial analogues of topological fixed point theorems. Here we prove the PPA-completeness of two problems of radically different style. They are PPA-Circuit CNSS and PPA-Circuit Chevalley, related respectively to the Combinatorial Nullstellensatz and to the Chevalley-Warning Theorem over the two elements field GF(2). The input of these problems contain PPA-circuits which are arithmetic circuits with special symmetric properties that assure that the polynomials computed by them have always an even number of zeros. In the proof of the result we relate the multilinear degree of the polynomials to the parity of the maximal parse subcircuits that compute monomials with maximal multilinear degree, and we show that the maximal parse subcircuits of a PPA-circuit can be paired in polynomial time.
关键词：Chevalley-Warning Theorem; Combinatorail Nullstellensatz; Polynomial Parity Argument; arithmetic circuit