Resultants and Discriminants for Bivariate Tensor-Product Polynomials

Mantzaflaris, Angelos; Tsigaridas, Elias

doi:10.1145/3087604.3087646

Cited by 5 publications

(5 citation statements)

References 53 publications

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“…However, there are very few results about determinantal formulas for mixed multihomogeneous systems, that is, when the supports are not the same. We know such formulas for scaled multihomogeneous systems [22], that is when the supports are scaled copies of one of them, and for bivariate tensor-product polynomial systems [38,9]. In what follows, we use the Weyman complex to derive new formulas for families of mixed multilinear systems.…”

Section: Koszul-type Formulamentioning

confidence: 99%

Koszul-type determinantal formulas for families of mixed multilinear systems

Bender,

Faugère,

Mantzaflaris

et al. 2021

Preprint

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Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.

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Section: Koszul-type Formulamentioning

confidence: 99%

Koszul-type determinantal formulas for families of mixed multilinear systems

Bender,

Faugère,

Mantzaflaris

et al. 2021

Preprint

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“…The map δ 1 (f 0 , m) corresponds to a Koszul-type formula, involving multiplication and dual multiplication maps. The matrix that represents this map is a Koszul resultant matrix [MT17,BMT17].…”

Section: Construction Of the Map δmentioning

confidence: 99%

“…Another kind of formula is the Koszul-type formula that involves the other maps of the Koszul complex. We call the matrices related to this formula Koszul resultant matri-ces [MT17,BMT17]. For both formulas, the elements of the matrices are linear polynomials in the coefficients of (f 0 , .…”

Section: Introductionmentioning

confidence: 99%

“…For unmixed multihomogeneous systems, that is systems where all the polynomial share the same support, determinantal formulas are well studied, e.g., [SZ94, WZ94, KS97, CK00, DD01, Wey03, DE03]. On the other hand, when we consider polynomials with different supports, that is mixed systems, little is known about determinantal formulas; with the exception of scaled multihomogeneous systems [EM12], that is when the supports are scaled copies of one of them, and the bivariate tensor-product case [MT17,BMT17].…”

Section: Introductionmentioning

confidence: 99%

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Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors

Bender,

Faugère,

Mantzaflaris

et al. 2018

Preprint

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A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that * This work is based on a paper originally published in ISSAC '18, July 16-19, 2018, New York, USA [BFMT18] is for systems consisting of multihomogeneous polynomials with the same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports.We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.

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“…A preliminary version of this paper appeared in [32]. In the current final version the Sections 4 and 5 are expanded significantly, and in particular the computation of (mixed) discriminants is improved by relating them with mixed resultants of lower degree.…”

Section: Introductionmentioning

confidence: 99%

Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomials

Busé

Mantzaflaris

Tsigaridas

2020

Journal of Symbolic Computation

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The construction of optimal resultant formulae for polynomial systems is one of the main areas of research in computational algebraic geometry. However, most of the constructions are restricted to formulae for unmixed polynomial systems, that is, systems of polynomials which all have the same support. Such a condition is restrictive, since mixed systems of equations arise frequently in many problems. Nevertheless, resultant formulae for mixed polynomial systems is a very challenging problem. We present a square, Koszul-type, matrix, the determinant of which is the resultant of an arbitrary (mixed) bivariate tensor-product polynomial system. The formula generalizes the classical Sylvester matrix of two univariate polynomials, since it expresses a map of degree one, that is, the elements of the corresponding matrix are up to sign the coefficients of the input polynomials. Interestingly, the matrix expresses a primal-dual multiplication map, that is, the tensor product of a univariate multiplication map with a map expressing derivation in a dual space. In addition we prove an impossibility result which states that for tensor-product systems with more than two (affine) variables there are no universal degree-one formulae, unless the system is unmixed. Last but not least, we present applications of the new construction in the efficient computation of discriminants and mixed discriminants.

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Resultants and Discriminants for Bivariate Tensor-Product Polynomials

Cited by 5 publications

References 53 publications

Koszul-type determinantal formulas for families of mixed multilinear systems

Koszul-type determinantal formulas for families of mixed multilinear systems

Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors

Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomials

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