TY - GEN

T1 - On the relation between polynomial identity testing and finding variable disjoint factors

AU - Shpilka, Amir

AU - Volkovich, Ilya

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=77955315480&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-14165-2_35

DO - 10.1007/978-3-642-14165-2_35

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AN - SCOPUS:77955315480

SN - 3642141641

SN - 9783642141645

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 408

EP - 419

BT - Automata, Languages and Programming - 37th International Colloquium, ICALP 2010, Proceedings

Y2 - 6 July 2010 through 10 July 2010

ER -