Multihomogeneous resultant matrices
Multihomogeneous structure in algebraic systems is the first step away from the classical theory of homogeneous equations towards fully exploiting arbitrary supports. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to p...
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2002
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS20461_v_n_p46_Dickenstein http://hdl.handle.net/20.500.12110/paper_NIS20461_v_n_p46_Dickenstein |
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paper:paper_NIS20461_v_n_p46_Dickenstein2023-06-08T16:39:57Z Multihomogeneous resultant matrices Degree vector Determinantal formula Multihomogeneous system Sparse resultant Sylvester and Bézout type matrix Algorithms Combinatorial mathematics Computer aided software engineering Computer simulation Game theory Polynomials Bezout types Classical theory of homogeneous equations Degree vector Determinantal formula MAPLE implementation Multihomogeneous resultant matrices Sparse resultant Sylvester type matrices Matrix algebra Multihomogeneous structure in algebraic systems is the first step away from the classical theory of homogeneous equations towards fully exploiting arbitrary supports. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, including hybrid matrices. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman [15, 18]. One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bézout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with an example showing all kinds of matrices that may be encountered, and illustrations of our MAPLE implementation. 2002 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS20461_v_n_p46_Dickenstein http://hdl.handle.net/20.500.12110/paper_NIS20461_v_n_p46_Dickenstein |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
| repository_str |
R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Degree vector Determinantal formula Multihomogeneous system Sparse resultant Sylvester and Bézout type matrix Algorithms Combinatorial mathematics Computer aided software engineering Computer simulation Game theory Polynomials Bezout types Classical theory of homogeneous equations Degree vector Determinantal formula MAPLE implementation Multihomogeneous resultant matrices Sparse resultant Sylvester type matrices Matrix algebra |
| spellingShingle |
Degree vector Determinantal formula Multihomogeneous system Sparse resultant Sylvester and Bézout type matrix Algorithms Combinatorial mathematics Computer aided software engineering Computer simulation Game theory Polynomials Bezout types Classical theory of homogeneous equations Degree vector Determinantal formula MAPLE implementation Multihomogeneous resultant matrices Sparse resultant Sylvester type matrices Matrix algebra Multihomogeneous resultant matrices |
| topic_facet |
Degree vector Determinantal formula Multihomogeneous system Sparse resultant Sylvester and Bézout type matrix Algorithms Combinatorial mathematics Computer aided software engineering Computer simulation Game theory Polynomials Bezout types Classical theory of homogeneous equations Degree vector Determinantal formula MAPLE implementation Multihomogeneous resultant matrices Sparse resultant Sylvester type matrices Matrix algebra |
| description |
Multihomogeneous structure in algebraic systems is the first step away from the classical theory of homogeneous equations towards fully exploiting arbitrary supports. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bézout types, including hybrid matrices. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman [15, 18]. One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bézout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with an example showing all kinds of matrices that may be encountered, and illustrations of our MAPLE implementation. |
| title |
Multihomogeneous resultant matrices |
| title_short |
Multihomogeneous resultant matrices |
| title_full |
Multihomogeneous resultant matrices |
| title_fullStr |
Multihomogeneous resultant matrices |
| title_full_unstemmed |
Multihomogeneous resultant matrices |
| title_sort |
multihomogeneous resultant matrices |
| publishDate |
2002 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_NIS20461_v_n_p46_Dickenstein http://hdl.handle.net/20.500.12110/paper_NIS20461_v_n_p46_Dickenstein |
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1769175832025104384 |