On the Relation between Polynomial Identity Testing and Finding Variable Disjoint Factors

We say that a polynomial f(x1, . . . , xn) 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 ...

• Type: Conference
• Year: 2010
• Where: ICALP
• Authors: Amir Shpilka, Ilya Volkovich
• Keywords: ICALP 2010, Polynomial, Polynomial Decomposition, Polynomial Identity Testing, Programming Languages