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

TL;DR

This paper introduces a novel approach to analyzing sparse curve singularities and their topological types using algebraic and tropical geometry techniques, including new results on Schur polynomials and ultratropicalization.

Contribution

It provides a new method for computing the delta-invariant of sparse curve singularities and describes the topological types of singularities of sparse resultants and algebraic knot diagrams.

Findings

Computed delta-invariant for generic sparse polynomial parameterized curves

Described topological types of singularities in sparse resultants

Introduced ultratropicalization as a refinement of tropicalization

Abstract

We compute the $\delta$-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.