On the relation between polynomial identity testing and finding variable disjoint factors

Amir Shpilka*, Ilya Volkovich

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

39 Scopus citations

Abstract

We say that a polynomial f(x 1,...,x n ) is indecomposable if it cannot be written as a product of two polynomials that are defined over disjoint sets of variables. The polynomial decomposition problem is defined to be the task of finding the indecomposable factors of a given polynomial. Note that for multilinear polynomials, factorization is the same as decomposition, as any two different factors are variable disjoint. In this paper we show that the problem of derandomizing polynomial identity testing is essentially equivalent to the problem of derandomizing algorithms for polynomial decomposition. More accurately, we show that for any reasonable circuit class there is a deterministic polynomial time (black-box) algorithm for polynomial identity testing of that class if and only if there is a deterministic polynomial time (black-box) algorithm for factoring a polynomial, computed in the class, to its indecomposable components. An immediate corollary is that polynomial identity testing and polynomial factorization are equivalent (up to a polynomial overhead) for multilinear polynomials. In addition, we observe that derandomizing the polynomial decomposition problem is equivalent, in the sense of Kabanets and Impagliazzo [1], to proving arithmetic circuit lower bounds for NEXP. Our approach uses ideas from [2], that showed that the polynomial identity testing problem for a circuit class is essentially equivalent to the problem of deciding whether a circuit from computes a polynomial that has a read-once arithmetic formula.

Original languageEnglish
Title of host publicationAutomata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings
Pages408-419
Number of pages12
EditionPART 1
DOIs
StatePublished - 2010
Externally publishedYes
Event37th International Colloquium on Automata, Languages and Programming, ICALP 2010 - Bordeaux, France
Duration: 6 Jul 201010 Jul 2010

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume6198 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference37th International Colloquium on Automata, Languages and Programming, ICALP 2010
Country/TerritoryFrance
CityBordeaux
Period6/07/1010/07/10

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