TY - JOUR

T1 - Equivalence of Polynomial Identity Testing and Polynomial Factorization

AU - Kopparty, Swastik

AU - Saraf, Shubhangi

AU - Shpilka, Amir

N1 - Publisher Copyright:
© 2015, Springer Basel.

PY - 2015/6/26

Y1 - 2015/6/26

N2 - In this paper, we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a multivariate polynomial f, the task of computing arithmetic circuits for the factors of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation of f). Together with the easy observation that deterministic factoring implies a deterministic algorithm for polynomial identity testing, this establishes an equivalence between these two central derandomization problems of arithmetic complexity. Previously, such an equivalence was known only for multilinear circuits (Shpilka & Volkovich, 2010).

AB - In this paper, we show that the problem of deterministically factoring multivariate polynomials reduces to the problem of deterministic polynomial identity testing. Specifically, we show that given an arithmetic circuit (either explicitly or via black-box access) that computes a multivariate polynomial f, the task of computing arithmetic circuits for the factors of f can be solved deterministically, given a deterministic algorithm for the polynomial identity testing problem (we require either a white-box or a black-box algorithm, depending on the representation of f). Together with the easy observation that deterministic factoring implies a deterministic algorithm for polynomial identity testing, this establishes an equivalence between these two central derandomization problems of arithmetic complexity. Previously, such an equivalence was known only for multilinear circuits (Shpilka & Volkovich, 2010).

KW - Arithmetic circuits

KW - polynomial factorization

KW - polynomial identity testing

UR - http://www.scopus.com/inward/record.url?scp=84929710007&partnerID=8YFLogxK

U2 - 10.1007/s00037-015-0102-y

DO - 10.1007/s00037-015-0102-y

M3 - מאמר

AN - SCOPUS:84929710007

VL - 24

SP - 295

EP - 331

JO - Computational Complexity

JF - Computational Complexity

SN - 1016-3328

IS - 2

M1 - 102

ER -