🤖 AI Summary
This work addresses the derandomization of Polynomial Identity Testing (PIT). We introduce a novel hitting-set generator based on low-degree univariate rational function evaluation, leveraging coordinate-wise variable association for efficient distinguishability. First, we establish scaling equivalence between rational-function-based hitting sets and the Shpilka–Volkovich generator. Second, we systematically characterize their vanishing ideal, deriving tight bounds on minimal degree, sparsity, and multilinear partition complexity. Third, we design a deterministic membership test via alternating algebra. Our framework uniformly reconstructs classical PIT derandomization results and—crucially—extends them to read-once oblivious algebraic branching programs (ROABPs), achieving both efficient derandomization and new lower-bound proofs. The approach bridges rational-function interpolation, algebraic geometry, and structural complexity, yielding improved conceptual clarity and broader applicability in arithmetic circuit complexity.
📝 Abstract
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ewcommand{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.