Recursive properties of the characteristic polynomial of weighted lattices

TL;DR

This paper investigates the recursive structure of the characteristic polynomial of weighted lattices, applying it to derive bounds on the critical exponent of q-polymatroids and analyzing their properties.

Contribution

It introduces a recursive description of the characteristic polynomial of weighted lattices and applies it to establish bounds and properties of q-polymatroids.

Findings

Recursive description of the characteristic polynomial of weighted lattices

A Critical Theorem for representable q-polymatroids

Lower bounds on the critical exponent of q-polymatroids

Abstract

In this paper, we describe properties of the characteristic polynomial of a weighted lattice and show that it has a recursive description, which we use to obtain results on the critical exponent of $q$-polymatroids. We give a Critical Theorem for representable $q$-polymatroids and we provide a lower bound on the critical exponent. We show that $q$-polymatroids arising from certain families of rank-metric codes attain this lower bound.