Valuative invariants for polymatroids

TL;DR

This paper constructs and analyzes modules of valuative invariants for matroids and polymatroids, confirming the universality of a key invariant and providing explicit bases and formulas.

Contribution

It develops the structure of valuative invariants modules, provides explicit bases, and proves the universality of the invariant $ ext{ extbackslash G}$ for these invariants.

Findings

Constructed $ ext{ extbackslash Z}$-modules of all valuative functions for matroids and polymatroids.

Provided explicit bases and formulas for ranks of these modules.

Confirmed the universality of the invariant $ ext{ extbackslash G}$.

Abstract

Many important invariants for matroids and polymatroids, such as the Tutte polynomial, the Billera-Jia-Reiner quasi-symmetric function, and the invariant $\mathcal G$ introduced by the first author, are valuative. In this paper we construct the $\Z$-modules of all $\Z$-valued valuative functions for labeled matroids and polymatroids on a fixed ground set, and their unlabeled counterparts, the $\Z$-modules of valuative invariants. We give explicit bases for these modules and for their dual modules generated by indicator functions of polytopes, and explicit formulas for their ranks. Our results confirm a conjecture of the first author that $\mathcal G$ is universal for valuative invariants.