# Jump loci for the rank of matrices and Betti numbers of chain complexes over Laurent polynomial rings

## TL;DR

This paper studies the structure of jump loci related to matrix ranks and Betti numbers of chain complexes over Laurent polynomial rings, generalizing previous results and including McCoy rank analysis.

## Contribution

It introduces a unified framework for analyzing jump loci for matrices and chain complexes over Laurent polynomial rings, extending prior work and encompassing McCoy rank.

## Key findings

- Characterization of jump loci for matrix ranks and Betti numbers
- Generalization of Kohno and Pajitnov's results on jump loci
- Inclusion of McCoy rank analysis in the framework

## Abstract

Let $K$ be a non-empty set of ideals of the commutative ring $R$, closed under taking smaller ideals. A subset $X$ of the group ring $R[\mathbb{Z}^s]$ is called a $K$-set if the ideal generated by the coefficients of the elements of $X$ is in $K$. For $X$ not a $K$-set we investigate the set of those homomorphisms $p \colon \mathbb{Z}^s \to \mathbb{Z}^t$ such that $p_*(X)$ is a $K$-set. We also consider corresponding notions of rank of matrices and Betti numbers of chain complexes; this includes an analysis of the case of McCoy rank. Our setup also recovers results on jump loci obtained by Kohno and Pajitnov as a special case.