Distinct matroid base weights and additive theory

Abstract

Let M be a matroid on a set E and let w: E → 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 formula for the number of distinct base weights. If | G | is a prime, our result coincides with a result of Schrijver and Seymour. We also describe Equality cases in this formula. In the prime case, our result generalizes Vosper's Theorem. © 2010 Springer Science+Business Media, LLC.