Computing Sparse Multiples of Polynomials

Giesbrecht, Mark; Roche, Daniel S.; Tilak, Hrushikesh

doi:10.1007/978-3-642-17517-6_25

Cited by 7 publications

(4 citation statements)

References 22 publications

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“…We ask for the shortest polynomial vanishing on X, or algebraically, the shortest polynomial in an ideal of the polynomial ring. The shortest polynomials contained in (principal) ideals of a univariate polynomial ring have been considered in [2]. Computing the shortest polynomials of an ideal in a polynomial ring seems to be a hard problem with an arithmetic flavor.…”

Section: Introductionmentioning

confidence: 99%

Short polynomials in determinantal ideals

Kahle,

Wiersig

2021

Preprint

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Section: Introductionmentioning

confidence: 99%

Short polynomials in determinantal ideals

Kahle,

Wiersig

2021

Preprint

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“…Open questions and avenues for future research are discussed in Section 6. An extended abstract of some of this work appears in Giesbrecht, Roche, and Tilak (2010). Some of this work and further explorations, also appears in the Masters thesis of Tilak (2010).…”

Section: Introductionmentioning

confidence: 97%

Computing Sparse Multiples of Polynomials

2012

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We consider the problem of finding a sparse multiple of a polynomial. Given f ∈ F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h ∈ F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F = Q and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t = 2.

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“…We ask for the shortest polynomial vanishing on

X

, or algebraically, the shortest polynomial in an ideal of the polynomial ring. The shortest polynomials contained in (principal) ideals of a univariate polynomial ring have been considered in [6]. Computing the shortest polynomials of an ideal in a polynomial ring seems to be a hard problem with an arithmetic flavour.…”

Section: Introductionmentioning

confidence: 99%

No short polynomials vanish on bounded rank matrices

Draisma

Kahle²,

Wiersig

2023

Bulletin of London Math Soc

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We show that the shortest non-zero polynomials vanishing on bounded-rank matrices and skew-symmetric matrices are the determinants and Pfaffians characterising the rank. Algebraically, this means that in the ideal generated by all 𝑡-minors or 𝑡-Pfaffians of a generic matrix or skew-symmetric matrix, one cannot find any polynomial with fewer terms than those determinants or Pfaffians, respectively, and that those determinants and Pfaffians are essentially the only polynomials in the ideal with that many terms. As a key tool of independent interest, we show that the ideal of a very general 𝑡-dimensional subspace of an affine 𝑛-space does not contain polynomials with fewer than 𝑡 + 1 terms.

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Computing Sparse Multiples of Polynomials

Cited by 7 publications

References 22 publications

Short polynomials in determinantal ideals

Short polynomials in determinantal ideals

Computing Sparse Multiples of Polynomials

No short polynomials vanish on bounded rank matrices

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