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Polynomial Identity Testing via Evaluation of Rational Functions

Authors Dieter van Melkebeek, Andrew Morgan

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Author Details

Dieter van Melkebeek
  • University of Wisconsin-Madison, Madison, WI, USA
Andrew Morgan
  • University of Wisconsin-Madison, Madison, WI, USA

Funding

Research partially supported by the US National Science Foundation through grant CCF-1838434.

Acknowledgements

We are indebted to Gautam Prakriya for helpful discussions and detailed feedback. We also thank Amir Shpilka for helpful comments and suggestions, as well as Ilya Volkovich and the anonymous reviewers. We are grateful to Hervé Fournier and Arpita Korwar for their presentation at WACT'18 in Paris [Fournier and Korwar, 2018].

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Dieter van Melkebeek and Andrew Morgan. Polynomial Identity Testing via Evaluation of Rational Functions. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 119:1-119:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ITCS.2022.119

Abstract

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. In spite of 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 size of set-multi-linearity 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 arithmetic branching programs that provably transcends the usual combinatorial techniques.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Derandomization
  • Gröbner Basis
  • Lower Bounds
  • Polynomial Identity Testing

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