Double Schubert polynomials do have saturated Newton polytopes

TL;DR

This paper proves that double Schubert polynomials possess the Saturated Newton Polytope property, confirming a conjecture and connecting algebraic geometry with combinatorial structures like discrete polymatroids.

Contribution

It establishes the Saturated Newton Polytope property for double Schubert polynomials, introducing a standardization of ideals to analyze multidegrees in non-standard multigradings.

Findings

Double Schubert polynomials have saturated Newton polytopes.

Support of multidegree polynomials forms a discrete polymatroid.

The approach links algebraic geometry with combinatorial structures.

Abstract

We prove that double Schubert polynomials have the Saturated Newton Polytope property. This settles a conjecture by Monical, Tokcan and Yong. Our ideas are motivated by the theory of multidegrees. We introduce a notion of standardization of ideals that enables us to study non-standard multigradings. This allows us to show that the support of the multidegree polynomial of each Cohen-Macaulay prime ideal, and in particular, that of each Schubert determinantal ideal is a discrete polymatroid.