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Abstract:

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.

Registro:

Documento: Artículo
Título:On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
Autor:Herrero, M.I.; Jeronimo, G.; Sabia, J.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, Argentina
Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS), CONICET-Universidad de Buenos Aires, Buenos Aires, Argentina
Ciclo Básico Común, Departamento de Ciencias Exactas, Universidad de Buenos Aires, Buenos Aires, Argentina
Palabras clave:Mixed volumes and mixed integrals; Multiplicity of zeros; Newton polytopes; Sparse polynomial systems
Año:2018
DOI: http://dx.doi.org/10.1007/s00454-018-0025-x
Título revista:Discrete and Computational Geometry
Título revista abreviado:Discrete Comput. Geom.
ISSN:01795376
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_01795376_v_n_p_Herrero

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Citas:

---------- APA ----------
Herrero, M.I., Jeronimo, G. & Sabia, J. (2018) . On the Multiplicity of Isolated Roots of Sparse Polynomial Systems. Discrete and Computational Geometry.
http://dx.doi.org/10.1007/s00454-018-0025-x
---------- CHICAGO ----------
Herrero, M.I., Jeronimo, G., Sabia, J. "On the Multiplicity of Isolated Roots of Sparse Polynomial Systems" . Discrete and Computational Geometry (2018).
http://dx.doi.org/10.1007/s00454-018-0025-x
---------- MLA ----------
Herrero, M.I., Jeronimo, G., Sabia, J. "On the Multiplicity of Isolated Roots of Sparse Polynomial Systems" . Discrete and Computational Geometry, 2018.
http://dx.doi.org/10.1007/s00454-018-0025-x
---------- VANCOUVER ----------
Herrero, M.I., Jeronimo, G., Sabia, J. On the Multiplicity of Isolated Roots of Sparse Polynomial Systems. Discrete Comput. Geom. 2018.
http://dx.doi.org/10.1007/s00454-018-0025-x