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Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness

Authors Markus Bläser, Anurag Pandey

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Author Details

Markus Bläser
  • Department of Computer Science, Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Anurag Pandey
  • Max Planck Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Pandey, Anurag: Supported by the Chair of Raimund Seidel, Department of Computer Science, Saarland University, Saarbrücken, Germany.

Acknowledgements

We thank Rohit Gurjar, Mrinal Kumar and Raimund Seidel for insightful discussions. We thank the Simons Institute for the Theory of Computing (Berkeley) and Schloss Dagstuhl - Leibniz-Zentrum für Informatik (Dagstuhl), for hosting us during certain phases of this research.

Cite As Get BibTex

Markus Bläser and Anurag Pandey. Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.8

Abstract

We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting.
Our algorithm works for every field 𝔽 with | 𝔽 | ≥ d+1, and uses only d log n + log (1/ ε) + O(d log log n) random bits to achieve a success probability 1 - ε for some ε > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity theory
  • Polynomial Identity Testing
  • Hitting Set
  • Pseudorandomness

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