Hybrid sparse resultant matrices for bivariate systems

Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sy...

Descripción completa

Guardado en:

Detalles Bibliográficos
Autores principales:	D'Andrea, C., Emiris, I.Z.
Formato:	CONF
Materias:	Convexity Mixed subdivision Piecewise-linear lifting Sparse resultant matrix Toric Jacobian Algorithms Boundary conditions Combinatorial mathematics Computer aided design Computer simulation Eigenvalues and eigenfunctions Geometry Piecewise linear techniques Polynomials Vectors Bivariate systems Newton polygons Piecewise linear lifting Matrix algebra
Acceso en línea:	http://hdl.handle.net/20.500.12110/paper_NIS02119_v_n_p24_DAndrea
Aporte de:	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires

id	todo:paper_NIS02119_v_n_p24_DAndrea
record_format	dspace
spelling	todo:paper_NIS02119_v_n_p24_DAndrea2023-10-03T16:45:44Z Hybrid sparse resultant matrices for bivariate systems D'Andrea, C. Emiris, I.Z. Convexity Mixed subdivision Piecewise-linear lifting Sparse resultant matrix Toric Jacobian Algorithms Boundary conditions Combinatorial mathematics Computer aided design Computer simulation Eigenvalues and eigenfunctions Geometry Piecewise linear techniques Polynomials Vectors Bivariate systems Mixed subdivision Newton polygons Piecewise linear lifting Sparse resultant matrix Toric Jacobian Matrix algebra Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our Maple implementation, applied to certain examples, illustrates our construction and compares with alternative matrices. Fil:D'Andrea, C. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. CONF info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_NIS02119_v_n_p24_DAndrea
institution	Universidad de Buenos Aires
institution_str	I-28
repository_str	R-134
collection	Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic	Convexity Mixed subdivision Piecewise-linear lifting Sparse resultant matrix Toric Jacobian Algorithms Boundary conditions Combinatorial mathematics Computer aided design Computer simulation Eigenvalues and eigenfunctions Geometry Piecewise linear techniques Polynomials Vectors Bivariate systems Mixed subdivision Newton polygons Piecewise linear lifting Sparse resultant matrix Toric Jacobian Matrix algebra
spellingShingle	Convexity Mixed subdivision Piecewise-linear lifting Sparse resultant matrix Toric Jacobian Algorithms Boundary conditions Combinatorial mathematics Computer aided design Computer simulation Eigenvalues and eigenfunctions Geometry Piecewise linear techniques Polynomials Vectors Bivariate systems Mixed subdivision Newton polygons Piecewise linear lifting Sparse resultant matrix Toric Jacobian Matrix algebra D'Andrea, C. Emiris, I.Z. Hybrid sparse resultant matrices for bivariate systems
topic_facet	Convexity Mixed subdivision Piecewise-linear lifting Sparse resultant matrix Toric Jacobian Algorithms Boundary conditions Combinatorial mathematics Computer aided design Computer simulation Eigenvalues and eigenfunctions Geometry Piecewise linear techniques Polynomials Vectors Bivariate systems Mixed subdivision Newton polygons Piecewise linear lifting Sparse resultant matrix Toric Jacobian Matrix algebra
description	Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our Maple implementation, applied to certain examples, illustrates our construction and compares with alternative matrices.
format	CONF
author	D'Andrea, C. Emiris, I.Z.
author_facet	D'Andrea, C. Emiris, I.Z.
author_sort	D'Andrea, C.
title	Hybrid sparse resultant matrices for bivariate systems
title_short	Hybrid sparse resultant matrices for bivariate systems
title_full	Hybrid sparse resultant matrices for bivariate systems
title_fullStr	Hybrid sparse resultant matrices for bivariate systems
title_full_unstemmed	Hybrid sparse resultant matrices for bivariate systems
title_sort	hybrid sparse resultant matrices for bivariate systems
url	http://hdl.handle.net/20.500.12110/paper_NIS02119_v_n_p24_DAndrea
work_keys_str_mv	AT dandreac hybridsparseresultantmatricesforbivariatesystems AT emirisiz hybridsparseresultantmatricesforbivariatesystems
_version_	1807319745045725184

Hybrid sparse resultant matrices for bivariate systems

Ejemplares similares