On Solving Sparse Polynomial Factorization Related Problems

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...

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Published in:Computational complexity Vol. 34; no. 1; p. 7
Main Authors: Bisht, Pranav, Volkovich, Ilya
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.06.2025
Springer Nature B.V
Subjects:
Algebra
Algorithm Analysis and Problem Complexity
Algorithms
Circuits
Computational Mathematics and Numerical Analysis
Computer Science
Polynomials
Sparsity
Multivariate Polynomial Factorization
Identity Testing
Pseudorandomness and derandomization
Factor Sparsity
Theory of computation- Algebraic complexity theory
Sparse Polynomials
Derandomization
ISSN:1016-3328, 1420-8954
Online Access:Get full text
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Summary: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 ( d 2 log 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 ] Π Σ Π [ ind - deg d ] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.
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ISSN:1016-3328
1420-8954
DOI:10.1007/s00037-025-00268-5