Uniform determinantal representations

A. Boralevi, M.J. van Doornmalen, J. Draisma, M.E. Hochstenbach, B. Plestenjak

Onderzoeksoutput: Bijdrage aan tijdschriftTijdschriftartikelAcademicpeer review

6 Citaten (Scopus)

Samenvatting


The problem of expressing a specific polynomial as the determinant of a square matrix of affine-linear forms arises from algebraic geometry, optimization, complexity theory, and scientific computing. Motivated by recent developments in this last area, we introduce the notion of a uniform determinantal representation, not of a single polynomial but rather of all polynomials in a given number of variables and of a given maximal degree. We derive a lower bound on the size of the matrix, and present a construction achieving that lower bound up to a constant factor as the number of variables is fixed and the degree grows. This construction marks an improvement upon a recent construction due to Plestenjak and Hochstenbach, and we investigate the performance of new representations in their root-finding technique for bivariate systems. Furthermore, we relate uniform determinantal representations to vector spaces of singular matrices, and we conclude with a number of future research directions.
Originele taal-2Engels
Pagina's (van-tot)415-441
Aantal pagina's27
TijdschriftSIAM Journal on Applied Algebra and Geometry
Volume1
Nummer van het tijdschrift1
DOI's
StatusGepubliceerd - 3 aug 2017

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