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On Solving Sparse Polynomial Factorization Related Problems

Authors Pranav Bisht, Ilya Volkovich

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Author Details

Pranav Bisht
  • Computer Science Department, Boston College, Chestnut Hill, MA, USA
Ilya Volkovich
  • Computer Science Department, Boston College, Chestnut Hill, MA, USA

Acknowledgements

The authors would like to thank the anonymous referees for their detailed comments and suggestions on the previous version of the paper.

Cite AsGet BibTex

Pranav Bisht and Ilya Volkovich. On Solving Sparse Polynomial Factorization Related Problems. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.FSTTCS.2022.10

Abstract

In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first factor sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most s terms and individual degree bounded by d can itself have at most s^O(d²log n) terms. It is conjectured, though, that the "true" sparsity bound should be polynomial (i.e. s^poly(d)). In this paper we provide supporting evidence for this conjecture by presenting polynomial-time algorithms for several problems that would be implied by a polynomial-size sparsity bound. In particular, we give efficient (deterministic) algorithms for identity testing of Σ^[2]ΠΣΠ^[ind-deg d] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Sparse Polynomials
  • Identity Testing
  • Derandomization
  • Factor-Sparsity
  • Multivariate Polynomial Factorization

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