  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.
