2025/10/07 by Andrew Goodall, Goodall, Andrew, Florent Jouve +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Meromorphic and Entire Functions #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2510.05658
openalex publication_date 2025/10/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.