Galois groups of multivariate Tutte polynomials

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.Comment: 8 pages, final version, to appear in J. Alg. Comb. Substantial revisions, including the addition of two alternative proofs of the main resul

Algebraic Properties of Chromatic Roots

“…Note that we have achieved every transitive permutation group of degree at most 4, but for degree 5 we are missing the cyclic group. The unique example of the dihedral group D 5 occurs for R (1,4,4,9,9,9,25). For degree 6, we have seen only five of the 16 transitive groups.…”

“…It was shown by de Mier et al [13] that, for a connected matroid (in particular, for a 2-connected graph), the two-variable Tutte polynomial is irreducible. Furthermore, Bohn et al [4] showed that, under the same hypotheses, the multivariate Tutte polynomial (regarded as a polynomial in the global variable over the field of fractions of all the local variables) has Galois group the symmetric group. Thus, one would expect that "almost all" specialisations of this polynomial would have symmetric Galois group.…”

Galois groups of chromatic polynomials

“…In most (over 90%) cases the Galois groups are symmetric Galois groups. Recently, it was shown that the Galois group of the multivariate Tutte-Whitney polynomial is always a direct product of symmetric groups [8]. The chromatic polynomial is an evaluation of this polynomial.…”