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On the irreducibility and monodromy of Tutte polynomials

Andrew Goodall, Florent Jouve, Jean-Sébastien Sereni

TL;DR

This work develops a unified framework for irreducibility and monodromy of Tutte polynomials across matroids and ranked sets. It introduces Brylawski polynomials, proves broad irreducibility criteria that apply to corank-nullity polynomials, and uses a probabilistic sieve to show that generic linear combinations of coprime Tutte polynomials have maximal Galois groups over $\mathbf{Q}(y)$. It then provides explicit monodromy computations for families such as cycle graphs, uniform matroids, and thick-edge cycles, yielding both maximal and non-maximal Galois behavior and proving BCM-type maximality for infinite families (e.g., cycles and uniform matroids). These results illuminate the algebro-geometric structure of Tutte polynomials and demonstrate how connectivity and rank-patterns influence Galois/monodromy behavior with potential implications for the broader landscape of polynomial matroid invariants.

Abstract

We study algebraic properties of the Tutte polynomial of a matroid and its generalizations to other combinatorially defined bivariate polynomial invariants. Merino, de Mier and Noy showed that the Tutte polynomial of a connected matroid is irreducible, and Bohn, Cameron and M{ü}ller conjectured the stronger property that the Galois/monodromy group of the Tutte polynomial of a connected matroid of rank r is isomorphic to the full symmetric group on r letters. First, we generalize the result of Merino-de Mier-Noy to the context of general ranked sets by exploiting a recent translation of the Brylawski relations, satisfied by the coefficients of the Tutte polynomial, into a functional identity. Second, we give the first confirmation of the conjecture of Bohn-Cameron-M{ü}ller for infinite families of connected matroids, including the cycle graphs and the uniform matroids. Moreover, we apply the large sieve to obtain a probabilistic statement showing that suitable linear combinations of coprime Tutte polynomials generically satisfy the conjecture.

On the irreducibility and monodromy of Tutte polynomials

TL;DR

This work develops a unified framework for irreducibility and monodromy of Tutte polynomials across matroids and ranked sets. It introduces Brylawski polynomials, proves broad irreducibility criteria that apply to corank-nullity polynomials, and uses a probabilistic sieve to show that generic linear combinations of coprime Tutte polynomials have maximal Galois groups over . It then provides explicit monodromy computations for families such as cycle graphs, uniform matroids, and thick-edge cycles, yielding both maximal and non-maximal Galois behavior and proving BCM-type maximality for infinite families (e.g., cycles and uniform matroids). These results illuminate the algebro-geometric structure of Tutte polynomials and demonstrate how connectivity and rank-patterns influence Galois/monodromy behavior with potential implications for the broader landscape of polynomial matroid invariants.

Abstract

We study algebraic properties of the Tutte polynomial of a matroid and its generalizations to other combinatorially defined bivariate polynomial invariants. Merino, de Mier and Noy showed that the Tutte polynomial of a connected matroid is irreducible, and Bohn, Cameron and M{ü}ller conjectured the stronger property that the Galois/monodromy group of the Tutte polynomial of a connected matroid of rank r is isomorphic to the full symmetric group on r letters. First, we generalize the result of Merino-de Mier-Noy to the context of general ranked sets by exploiting a recent translation of the Brylawski relations, satisfied by the coefficients of the Tutte polynomial, into a functional identity. Second, we give the first confirmation of the conjecture of Bohn-Cameron-M{ü}ller for infinite families of connected matroids, including the cycle graphs and the uniform matroids. Moreover, we apply the large sieve to obtain a probabilistic statement showing that suitable linear combinations of coprime Tutte polynomials generically satisfy the conjecture.

Paper Structure

This paper contains 16 sections, 27 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. The Tutte polynomial of a ranked set
  3. Brylawski polynomials
  4. Irreducibility of Brylawski polynomials
  5. Probabilistic approach: generic Galois maximality
  6. Generalizing a uniform version of a Theorem of Gallagher
  7. Application to linear combinations of Tutte polynomials of connected matroids
  8. Monodromy computations for some ranked sets and specific families of 2-connected graphs
  9. Examples of monodromy group of Tutte polynomials of ranked sets
  10. A rank function assuming two values
  11. A rank function assuming three values
  12. The case of an n-cycle
  13. The case of uniform matroids
  14. The case of an n-cycle with a "thick edge"
  15. The case where $j$ is odd
  16. ...and 1 more sections

Key Result

Proposition 2.1

For a ranked set $S=(E,\mathop{\mathrm{r}}\nolimits)$, In other words,

Theorems & Definitions (61)

  • Conjecture 1.1: Bohn--Cameron--Müller
  • Definition 1
  • Proposition 2.1
  • proof
  • Definition 2
  • Definition 3
  • Proposition 2.2
  • proof
  • Remark 1
  • Proposition 2.3: BCCP23*proof of Theorem 1.1
  • ...and 51 more