Koszul-type determinantal formulas for families of mixed multilinear systems - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles SIAM Journal on Applied Algebra and Geometry Year : 2021

Koszul-type determinantal formulas for families of mixed multilinear systems

Abstract

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.
Fichier principal
Vignette du fichier
2bilinear.pdf (496.73 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03236344 , version 1 (26-05-2021)

Identifiers

Cite

Matías R Bender, Jean-Charles Faugère, Angelos Mantzaflaris, Elias Tsigaridas. Koszul-type determinantal formulas for families of mixed multilinear systems. SIAM Journal on Applied Algebra and Geometry, 2021, 5 (4), pp.589-619. ⟨10.1137/20M1332190⟩. ⟨hal-03236344⟩
113 View
184 Download

Altmetric

Share

Gmail Mastodon Facebook X LinkedIn More