文章基本信息

标题：On the Complexity of Symmetric Polynomials
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作者：Markus Bl{"a}ser ; Gorav Jindal
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2018
卷号：124
页码：1-14
DOI：10.4230/LIPIcs.ITCS.2019.47
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：The fundamental theorem of symmetric polynomials states that for a symmetric polynomial f_{Sym} in C[x_1,x_2,...,x_n], there exists a unique "witness" f in C[y_1,y_2,...,y_n] such that f_{Sym}=f(e_1,e_2,...,e_n), where the e_i's are the elementary symmetric polynomials. In this paper, we study the arithmetic complexity L(f) of the witness f as a function of the arithmetic complexity L(f_{Sym}) of f_{Sym}. We show that the arithmetic complexity L(f) of f is bounded by poly(L(f_{Sym}),deg(f),n). To the best of our knowledge, prior to this work only exponential upper bounds were known for L(f). The main ingredient in our result is an algebraic analogue of Newton's iteration on power series. As a corollary of this result, we show that if VP != VNP then there exist symmetric polynomial families which have super-polynomial arithmetic complexity. Furthermore, we study the complexity of testing whether a function is symmetric. For polynomials, this question is equivalent to arithmetic circuit identity testing. In contrast to this, we show that it is hard for Boolean functions.
关键词：Symmetric Polynomials; Arithmetic Circuits; Arithmetic Complexity; Power Series; Elementary Symmetric Polynomials; Newton's Iteration