Supports of Castelnuovo-Mumford polynomials

TL;DR

This paper investigates the support of Castelnuovo-Mumford polynomials, showing that for certain permutation families, the support is M-convex and corresponds to integer points in a schubitope, advancing understanding of their geometric structure.

Contribution

The paper introduces new permutation families with Castelnuovo-Mumford polynomials whose supports are M-convex and correspond to schubitopes, extending known cases.

Findings

Support of certain Castelnuovo-Mumford polynomials is M-convex.

Supports correspond to integer points in schubitopes for specific permutations.

New families of permutations with M-convex polynomial support are identified.

Abstract

The Castelnuovo-Mumford polynomials are the maximal degree components of Grothendieck polynomials. The support of each Castelnuovo-Mumford polynomial is conjectured to be M-convex, i.e. the set of integer points of a generalized permutahedron (M\'esz\'aros and St. Dizier, 2020). This conjecture is known to hold in certain special cases but remains open in general. We define new families of permutations whose Castelnuovo-Mumford polynomials we show to have M-convex support. Specifically, we investigate which permutations have Castelnuovo-Mumford polynomials whose supports are the set of integer points in a schubitope.