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arXiv · 2402.08410

Sparse systems with high local multiplicity

Frédéric Bihan·Alicia Dickenstein·Jens Forsgård
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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.

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Frédéric Bihan, Alicia Dickenstein, Jens Forsgård. 2025-02-08. Sparse systems with high local multiplicity. https://arxiv.org/abs/2402.08410

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