Pré-Publication, Document De Travail Année : 2025

On the irreducibility and monodromy of Tutte polynomials

Résumé

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 Müller 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-Müller 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.

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Dates et versions

hal-05299433 , version 1 (06-10-2025)

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  • HAL Id : hal-05299433 , version 1

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Andrew Goodall, Florent Jouve, Jean-Sébastien Sereni. On the irreducibility and monodromy of Tutte polynomials. 2025. ⟨hal-05299433⟩
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