Distinct Matroid Base Weights and Additive Theory

TL;DR

This paper investigates the number of distinct weights of bases in a matroid with weights in a cyclic group, providing formulas under Pollard's Condition and generalizing known theorems like Vosper's.

Contribution

It derives explicit formulas for the number of distinct base weights in matroids with weights in cyclic groups, extending previous results and characterizing equality cases.

Findings

Formula for the number of distinct base weights under Pollard's Condition

Generalization of Vosper's Theorem for prime cyclic groups

Characterization of cases where the formula achieves equality

Abstract

Let $M$ be a matroid on a set $E$ and let $w:E\longrightarrow G$ be a weight function, where $G$ is a cyclic group. Assuming that $w(E)$ satisfies the Pollard's Condition (i.e. Every non-zero element of $w(E)-w(E)$ generates $G$), we obtain a formulae for the number of distinct base weights. If $|G|$ is a prime, our result coincides with a result Schrijver and Seymour. We also describe Equality cases in this formulae. In the prime case, our result generalizes Vosper's Theorem.