ECCC-Report TR14-001https://eccc.weizmann.ac.il/report/2014/001Comments and Revisions published for TR14-001en-usSat, 04 Jan 2014 10:47:05 +0200
Paper TR14-001
| Equivalence of Polynomial Identity Testing and Deterministic Multivariate Polynomial Factorization |
Swastik Kopparty,
Shubhangi Saraf,
Amir Shpilka
https://eccc.weizmann.ac.il/report/2014/001In 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 polynomial $f(X_1,\ldots,X_n)$, 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 and Volkovich, ICALP 2010).
Sat, 04 Jan 2014 10:47:05 +0200https://eccc.weizmann.ac.il/report/2014/001