Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sparse systems with high local multiplicity

Frédéric Bihan, Alicia Dickenstein, Jens Forsgård

Abstract

Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.

Sparse systems with high local multiplicity

Abstract

Consider a sparse system of n Laurent polynomials in n variables with complex coefficients and support in a finite lattice set A. The maximal number of isolated roots of the system in the complex n-torus is known to be the normalized volume of the convex hull of A (the BKK bound). We explore the following question: if the cardinality of A equals n+m+1, what is the maximum local intersection multiplicity at one point in the torus in terms of n and m? This study was initiated by Gabrielov in the multivariate case. We give an upper bound that is always sharp when m=1 and, under a generic technical hypothesis, it is considerably smaller for any dimension n and codimension m. We also present, for any value of n and m, a particular sparse system with high local multiplicity with exponents in the vertices of a cyclic polytope and we explain the rationale of our choice. Our work raises several interesting questions.
Paper Structure (13 sections, 41 theorems, 109 equations, 2 figures, 1 table)

This paper contains 13 sections, 41 theorems, 109 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multiplicities and mixed covolumes
  3. Bounds for the local intersection multiplicity
  4. Systems with integer coefficients
  5. Explicit bounds
  6. Consequences and improvements of Theorem \ref{['T:t_i']}
  7. Local Gale duality for polynomial systems
  8. $H$-duality
  9. Viewing all the systems together
  10. Bounds for the local intersection multiplicity via duality
  11. An explicit sparse system with high local multiplicity
  12. The codimension $1$ case
  13. Multiplicity of one point on one hypersurface

Key Result

Theorem 2.5

M, E, HJSMBook Let $F_{1},\ldots, F_{n} \in \mathbb{C}[x_{1},\ldots,x_{n}]$ be polynomials which vanish at the origin. If $F_{1},\ldots, F_{n}$ are convenient, then Moreover, if the polynomial system $F_{1}=\cdots=F_{n}=0$ is non-degenerate at the origin, then

Figures (2)

  • Figure 1: The Newton diagrams ${\mathcal{D}}(F_{1}), {\mathcal{D}}(F_{2})$ and their Minkowski sum, where $a= \frac{m+1}{2}$ ($m$ odd). We see that the mixed covolume equals $\operatorname{Vol}^{\circ}(\Delta_{1},\Delta_{2}) = a \cdot (2a+1)={m+2 \choose 2}$.
  • Figure 2: A local deformation of a real dessin d'enfant with a triple point (valency six) marked with a letter $r$ to a real dessin d'enfant with three simple points (valency two each) marked with a letter $r$. The horizontal segment represents part ot $\mathbb{R} P^{1}$.

Theorems & Definitions (100)

  • Definition 1.5
  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • Example 2.8
  • Theorem 3.1
  • ...and 90 more