info:eu-repo/semantics/article
On the Multiplicity of Isolated Roots of Sparse Polynomial Systems
Fecha
2019-12Registro en:
Herrero, Maria Isabel; Jeronimo, Gabriela Tali; Sabia, Juan Vicente Rafael; On the Multiplicity of Isolated Roots of Sparse Polynomial Systems; Springer; Discrete And Computational Geometry; 62; 4; 12-2019; 788-812
0179-5376
CONICET Digital
CONICET
Autor
Herrero, Maria Isabel
Jeronimo, Gabriela Tali
Sabia, Juan Vicente Rafael
Resumen
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.