Discrete & Computational Geometry

, Volume 62, Issue 4, pp 788–812

# On the Multiplicity of Isolated Roots of Sparse Polynomial Systems

Article

## 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.

## Keywords

Sparse polynomial systems Multiplicity of zeros Newton polytopes Mixed volumes and mixed integrals

## Mathematics Subject Classification

13H15 14Q99 14C17

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## Authors and Affiliations

• María Isabel Herrero
• 1
• 2
• Gabriela Jeronimo
• 1
• 2
• 3
• Juan Sabia
• 2
• 3
1. 1.Departamento de Matemática, Facultad de Ciencias Exactas y NaturalesUniversidad de Buenos AiresBuenos AiresArgentina
2. 2.Instituto de Investigaciones Matemáticas Luis A. Santaló (IMAS)CONICET-Universidad de Buenos AiresBuenos AiresArgentina
3. 3.Ciclo Básico Común, Departamento de Ciencias ExactasUniversidad de Buenos AiresBuenos AiresArgentina

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