4 research outputs found
Polynomial Identity Testing via Evaluation of Rational Functions
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. Despite the univariate nature, we establish an equivalence
up to rescaling with a generator introduced by Shpilka and Volkovich, which has
a similar structure but uses multivariate polynomials in the abscissas.
We study the power of the generator by characterizing its vanishing ideal,
i.e., the set of polynomials that it fails to hit. Capitalizing on the
univariate nature, 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. Inspired by an alternating algebra representation, we
develop a structured deterministic membership test for the vanishing ideal. As
a proof of concept, we rederive known derandomization results based on the
generator by Shpilka and Volkovich and present a new application for read-once
oblivious algebraic branching programs.Comment: Appeared at ITCS 202