核心概念
The paper introduces a hitting set generator for Polynomial Identity Testing based on evaluations of low-degree univariate rational functions, and provides a systematic characterization of its vanishing ideal, which enables new derandomization results and lower bounds.
要約
The paper introduces the Rational Function Evaluation (RFE) generator as an alternative view of the Shpilka-Volkovich (SV) generator for Polynomial Identity Testing (PIT). The RFE generator substitutes each variable xi with a low-degree univariate rational function f(ai), where the abscissas ai are distinct field elements.
The main contributions are:
- Characterization of the vanishing ideal of RFE:
- Theorem 3 provides a small and explicit generating set for the vanishing ideal, consisting of determinant expressions called "Elementary Vandermonde Circulations" (EVCs).
- Corollaries 5 and 6 derive tight bounds on the minimum degree and sparseness of polynomials in the vanishing ideal.
- Corollary 7 establishes the minimum partition class size of set-multilinear polynomials in the vanishing ideal.
- Structured membership test for multilinear polynomials in the vanishing ideal (Theorem 8):
- The test uses partial derivatives and zero substitutions, providing insight into why the generator hits certain polynomials.
- Applications:
- Theorem 9 shows that SVl hits read-once oblivious algebraic branching programs (ROABPs) of width less than (l/3) + 1 that contain a monomial of degree at most l + 1.
- Theorem 10 establishes corresponding lower bounds on ROABP width for polynomials in the vanishing ideal.
The paper also discusses connections to alternating algebra and network flow, which provide intuition for the results.
統計
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