# Sparse curve* *singularities

Algebraic geometry

Generic singularities of both the resultants of sparse polynomials and the curve projections given by sparse polynomials are described.

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

Submitted (2023)

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.

Submitted (2023)