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Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials

Authors Pranav Bisht , Nitin Saxena

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
  • Theory of computation → Pseudorandomness and derandomization
  • Mathematics of computing → Combinatoric problems
Keywords
  • Multivariate polynomial factorization
  • derandomization
  • sparse polynomials
  • symmetric polynomials
  • factor-sparsity
  • convex polytopes

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Abstract

More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s-sparse polynomials, having individual degree d = O(1), is very well-studied (Shpilka, Volkovich ICALP'10; Volkovich RANDOM'17; Bhargava, Saraf and Volkovich FOCS'18, JACM'20). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s^poly(d)-sparsity bound for the factors of symmetric polynomials over any field. This characterizes the known worst-case examples of sparsity blow-up for sparse polynomial factoring.
To factor f, we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f. We prove a crucial result about convex polytopes, by introducing the concept of "low min-entropy", which might also be of independent interest.

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Pranav Bisht and Nitin Saxena. Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials. 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. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.FSTTCS.2022.9

Author Details

Pranav Bisht
  • Department of Computer Science & Engineering, IIT Kanpur, India
Nitin Saxena
  • Department of Computer Science & Engineering, IIT Kanpur, India

Funding

  • Saxena, Nitin: DST-SERB (CRG/2020/000045) and N. Rama Rao Chair.

Acknowledgements

Pranav thanks Ilya Volkovich and Vishwas Bhargava for useful discussions. The authors would also like to thank the anonymous reviewers for their useful comments that improved the presentation of results.

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