Sparse curve singularities, singular loci of resultants, and Vandermonde matrices

2023/01/17 by Alexander Esterov, Esterov, Alexander, Evgeny Statnik +3
Mathematics · #14M25 #14N10 #32S25 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2301.07001

openalex publication_date 2023/01/17 · openalex created_date 2023/01/20 · openalex updated_date 2026/07/28

Abstract

We compute the δ-invariant of a curve singularity parameterized by generic sparse polynomials. We apply this to describe topological types of generic singularities of sparse resultants and ``algebraic knot diagrams'' (i.e. generic algebraic spatial curve projections). Our approach is based on some new results on zero loci of Schur polynomials, on transversality properties of maps defined by sparse polynomials, and on a new refinement of the notion of tropicalization of a curve (ultratropicalization), which may be of independent interest.

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