Galois groups of multivariate Tutte polynomials

TL;DR

This paper proves that the Galois group of the multivariate Tutte polynomial of a connected matroid over a field is the symmetric group, revealing deep algebraic symmetry properties of these polynomials.

Contribution

It establishes the irreducibility of the multivariate Tutte polynomial and identifies its Galois group as a symmetric group, providing new insights into its algebraic structure.

Findings

Galois group of multivariate Tutte polynomial is the symmetric group

Polynomial is irreducible over the field of rational functions in variables

Galois group of any matroid's Tutte polynomial is a product of symmetric groups

Abstract

The multivariate Tutte polynomial $\hat Z_M$ of a matroid $M$ is a generalization of the standard two-variable version, obtained by assigning a separate variable $v_e$ to each element $e$ of the ground set $E$. It encodes the full structure of $M$. Let $\bv = \{v_e\}_{e\in E}$, let $K$ be an arbitrary field, and suppose $M$ is connected. We show that $\hat Z_M$ is irreducible over $K(\bv)$, and give three self-contained proofs that the Galois group of $\hat Z_M$ over $K(\bv)$ is the symmetric group of degree $n$, where $n$ is the rank of $M$. An immediate consequence of this result is that the Galois group of the multivariate Tutte polynomial of any matroid is a direct product of symmetric groups. Finally, we conjecture a similar result for the standard Tutte polynomial of a connected matroid.