Matroid base polytope decomposition II : sequence of hyperplane splits

TL;DR

This paper investigates conditions under which matroid base polytopes can be decomposed into multiple parts via hyperplane splits, extending previous work and providing both sufficient and necessary conditions for such decompositions.

Contribution

It introduces new sufficient conditions for hyperplane split sequences in matroid base polytopes and explores decomposition properties for direct sums of matroids.

Findings

Infinite decompositions for many matroids

Necessary conditions for rank three matroids

Decomposition of direct sums based on component properties

Abstract

This is a continuation of the early paper concerning matroid base polytope decomposition. Here, we will present sufficient conditions on $M$ so its base matroid polytope $P(M)$ has a {\em sequence} of hyperplane splits. The latter yields to decompositions of $P(M)$ with two or more pieces for infinitely many matroids $M$. We also present necessary conditions on the Euclidean representation of rank three matroids $M$ for the existences of decompositions of $P(M)$ into $2$ or $3$ pieces. Finally, we prove that $P(M_1 \oplus M_2)$ has a sequence of hyperplane splits if either $P(M_1)$ or $P(M_2)$ also has a sequence of hyperplane splits.