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: | , |
| Format: | Journal Article |
| Language: | English |
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Springer International Publishing
01.06.2025
Springer Nature B.V |
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| ISSN: | 1016-3328, 1420-8954 |
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| 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
(
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. |
|---|---|
| AbstractList | 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 sO(d2logn) terms. It is conjectured, though, that the ``true'' sparsity bound should be polynomial (i.e., spoly(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-degd] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques. 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. |
| ArticleNumber | 7 |
| Author | Bisht, Pranav Volkovich, Ilya |
| Author_xml | – sequence: 1 givenname: Pranav surname: Bisht fullname: Bisht, Pranav email: pranav@iitism.ac.in organization: Department of Computer Science and Engineering , IIT(ISM) Dhanbad – sequence: 2 givenname: Ilya surname: Volkovich fullname: Volkovich, Ilya organization: Computer Science Department, Boston College |
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| Keywords | Multivariate Polynomial Factorization Identity Testing Pseudorandomness and derandomization Factor Sparsity Theory of computation- Algebraic complexity theory Sparse Polynomials Derandomization |
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| SubjectTerms | Algebra Algorithm Analysis and Problem Complexity Algorithms Circuits Computational Mathematics and Numerical Analysis Computer Science Polynomials Sparsity |
| Title | On Solving Sparse Polynomial Factorization Related Problems |
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