Tropicalization, symmetric polynomials, and complexity

TL;DR

This paper proves that tropical skew Schur polynomials have polynomial tropical semiring complexity, establishing an upper bound and connecting it to properties of Schubert polynomials and Newton polytopes.

Contribution

It provides the first polynomial complexity upper bound for tropical skew Schur polynomials, confirming a conjecture and linking complexity to geometric properties.

Findings

Tropical Schur polynomials have polynomial complexity.

Tropical skew Schur polynomials also have polynomial complexity.

Complexity bounds are connected to saturated Newton polytope property.

Abstract

D. Grigoriev-G. Koshevoy recently proved that tropical Schur polynomials have (at worst) polynomial tropical semiring complexity. They also conjectured tropical skew Schur polynomials have at least exponential complexity; we establish a polynomial complexity upper bound. Our proof uses results about (stable) Schubert polynomials, due to R. P. Stanley and S. Billey-W. Jockusch-R. P. Stanley, together with a sufficient condition for polynomial complexity that is connected to the saturated Newton polytope property.