Polynomial Identity Testing via Evaluation of Rational Functions

TL;DR

This paper introduces a new hitting set generator for Polynomial Identity Testing using evaluations of low-degree univariate rational functions, providing a systematic analytic framework and applications in derandomization and lower bounds.

Contribution

It develops a novel rational function-based generator, characterizes its vanishing ideal, and offers a structured deterministic membership test, advancing the analytic understanding of hitting set generators.

Findings

Established equivalence with Shpilka-Volkovich generator

Provided tight bounds on vanishing ideal properties

Rederived known derandomization results and introduced new applications

Abstract

$ \newcommand{\ie}{i.\,e.} $We introduce a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions at abscissas associated with the variables. We establish an equivalence up to rescaling with a generator introduced by Shpilka and Volkovich, which has a similar structure but uses multivariate polynomials. We initiate a systematic analytic study of the power of hitting set generators by characterizing their vanishing ideals, \ie, the sets of polynomials that they fail to hit. We provide two such characterizations for our generator. First, we develop a small collection of polynomials that jointly produce the vanishing ideal. As corollaries, we obtain tight bounds on the minimum degree, sparseness, and partition class size of set-multilinearity in the vanishing ideal. Second, inspired by a connection to alternating algebra, we develop a structured deterministic membership test for the multilinear part of the vanishing ideal. We present a derivation based on alternating algebra along with the required background, as well as one in terms of zero substitutions and partial derivatives, avoiding the need for alternating algebra. As evidence of the utility of our analytic approach, we rederive known derandomization results based on the generator by Shpilka and Volkovich and present a new application in derandomization / lower bounds for read-once oblivious algebraic branching programs.