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 torus ${\left(C^{\ast }\right)}^n$ 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 [13] in the multivariate case. We give an upper bound that is always sharp when $m=1$ and, under a technical hypothesis, it is considerably smaller than the previous upper bound 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.