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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| Veröffentlicht in: | Computational complexity Jg. 34; H. 1; S. 7 |
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| Sprache: | Englisch |
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Springer International Publishing
01.06.2025
Springer Nature B.V |
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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
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. 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. |
| ArticleNumber | 7 |
| Author | Bisht, Pranav Volkovich, Ilya |
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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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