On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its...
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2018
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v_n_p_Herrero http://hdl.handle.net/20.500.12110/paper_01795376_v_n_p_Herrero |
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paper:paper_01795376_v_n_p_Herrero2023-06-08T15:19:31Z On the Multiplicity of Isolated Roots of Sparse Polynomial Systems Mixed volumes and mixed integrals Multiplicity of zeros Newton polytopes Sparse polynomial systems We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions. © 2018, Springer Science+Business Media, LLC, part of Springer Nature. 2018 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v_n_p_Herrero http://hdl.handle.net/20.500.12110/paper_01795376_v_n_p_Herrero |
| institution |
Universidad de Buenos Aires |
| institution_str |
I-28 |
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R-134 |
| collection |
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) |
| topic |
Mixed volumes and mixed integrals Multiplicity of zeros Newton polytopes Sparse polynomial systems |
| spellingShingle |
Mixed volumes and mixed integrals Multiplicity of zeros Newton polytopes Sparse polynomial systems On the Multiplicity of Isolated Roots of Sparse Polynomial Systems |
| topic_facet |
Mixed volumes and mixed integrals Multiplicity of zeros Newton polytopes Sparse polynomial systems |
| description |
We give formulas for the multiplicity of any affine isolated zero of a generic polynomial system of n equations in n unknowns with prescribed sets of monomials. First, we consider sets of supports such that the origin is an isolated root of the corresponding generic system and prove formulas for its multiplicity. Then, we apply these formulas to solve the problem in the general case, by showing that the multiplicity of an arbitrary affine isolated zero of a generic system with given supports equals the multiplicity of the origin as a common zero of a generic system with an associated family of supports. The formulas obtained are in the spirit of the classical Bernstein’s theorem, in the sense that they depend on the combinatorial structure of the system, namely, geometric numerical invariants associated to the supports, such as mixed volumes of convex sets and, alternatively, mixed integrals of convex functions. © 2018, Springer Science+Business Media, LLC, part of Springer Nature. |
| title |
On the Multiplicity of Isolated Roots of Sparse Polynomial Systems |
| title_short |
On the Multiplicity of Isolated Roots of Sparse Polynomial Systems |
| title_full |
On the Multiplicity of Isolated Roots of Sparse Polynomial Systems |
| title_fullStr |
On the Multiplicity of Isolated Roots of Sparse Polynomial Systems |
| title_full_unstemmed |
On the Multiplicity of Isolated Roots of Sparse Polynomial Systems |
| title_sort |
on the multiplicity of isolated roots of sparse polynomial systems |
| publishDate |
2018 |
| url |
https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v_n_p_Herrero http://hdl.handle.net/20.500.12110/paper_01795376_v_n_p_Herrero |
| _version_ |
1768544362723016704 |