Intersections of base rings associated to transversal polymatroids

TL;DR

This paper investigates the conditions under which intersections of base rings associated with transversal polymatroids are Gorenstein, providing necessary and sufficient criteria and computing their $a$-invariants, supported by computer algebra experiments.

Contribution

It establishes when intersections of certain base rings remain base rings of transversal polymatroids and computes their $a$-invariants, extending previous work on Gorenstein properties.

Findings

Intersections of base rings are Gorenstein under specific conditions.

Necessary and sufficient conditions for the intersection to be a base ring of a transversal polymatroid.

Computed $a$-invariants for these base rings.

Abstract

The discrete polymatroids and their base rings are studied recently in many papers (see \cite{HH}, \cite{HHV}, \cite{V1}, \cite{V2}). It is important to give conditions when the base ring associated to a transversal polymatroid is Gorenstein (see \cite{HH}). In \cite{SA} we introduced a class of such base rings. In this paper we note that an intersection of such base rings (introduced in \cite{SA}) is Gorenstein and give necessary and sufficient conditions for the intersection of two base rings from \cite{SA} to be still a base ring of a transversal polymatroid. Also, we compute the $a$-invariant of those base rings. The results presented were discovered by extensive computer algebra experiments performed with {\it{Normaliz}} \cite{BK}.