# On zeros of the characteristic polynomial of matroids of bounded tree-width

## TL;DR

This paper establishes bounds on the real zeros of the characteristic polynomial for certain representable matroids with bounded tree-width, linking matroid structure to polynomial roots.

## Contribution

It introduces tools for analyzing representable matroids of bounded tree-width and proves a bound on the largest real zero of their characteristic polynomial.

## Key findings

- Characteristic polynomial zeros are bounded by q^{k-1} for matroids of tree-width k
- Tools developed for analyzing GF(q)-representable matroids
- Bounded tree-width constrains polynomial roots

## Abstract

We develop some basic tools to work with representable matroids of bounded tree-width and use them to prove that, for any prime power $q$ and constant $k$, the characteristic polynomial of any loopless, $GF(q)$-representable matroid with tree-width $k$ has no real zero greater than $q^{k-1}$.