Sparse systems with high local multiplicity
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 fol...
Saved in:
Main Authors | , , |
---|---|
Format | Journal Article |
Language | English |
Published |
13.02.2024
|
Subjects | |
Online Access | Get full text |
Cover
Loading…
Summary: | 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. |
---|---|
DOI: | 10.48550/arxiv.2402.08410 |