A. Esterov

E. Statnik

A. Voorhaar

Geometric properties, including delta invariants, are computed for singular points defined by polynomials with indeterminate coefficients.

Singularities, such as cusps or self-intersections, are points where mathematical objects such as functions or surfaces cease to be well-behaved. Extensively studied, important examples of these points can, intriguingly, be described by polynomials with indeterminate coefficients. We deduce some general results for singularities of this form, including a formula for the delta invariant, an index of their complexity.

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.

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