Sparse systems with high local multiplicity

Abstract Consider a sparse system of Laurent polynomials in variables with complex coefficients and support in a finite lattice set . The maximal number of isolated roots of the system in the torus is known to be the normalized volume of the convex hull of (the BKK bound). We explore the following question: if the cardinality of equals , what is the maximum local intersection multiplicity at one point in the torus in terms of and ? This study was initiated by Gabrielov [13] in the multivariate case. We give an upper bound that is always sharp when and, under a technical hypothesis, it is considerably smaller than the previous upper bound for any dimension and codimension . We also present, for any value of and , 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.