Related papers: Galois groups of multivariate Tutte polynomials

The variety of principal minors of $n\times n$ symmetric matrices, denoted $Z_{n}$, is invariant under the action of a group $G\subset \GL(2^{n})$ isomorphic to $\G$. We describe an irreducible $G$-module of degree $4$ polynomials…

Cut-and-paste $K$-theory is a new variant of higher algebraic $K$-theory that has proven to be useful in problems involving decompositions of combinatorial and geometric objects, e.g., scissors congruence of polyhedra and reconstruction…

The classical Tutte polynomial is a two-variate polynomial $T_G(x,y)$ associated to graphs or more generally, matroids. In this paper, we introduce a polynomial $\widetilde{T}_H(x,y)$ associated to a bipartite graph $H$ that we call the…

Delta-matroids are "type B" generalizations of matroids in the same way that maximal orthogonal Grassmannians are generalizations of Grassmannians. A delta-matroid analogue of the Tutte polynomial of a matroid is the interlace polynomial.…

In 1999 Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article we show that the conjecture generalized to matroids holds for the large class of all split matroids by…

In the present paper, we introduce the concept of harmonic Tutte polynomials of matroids and discuss some of their properties. In particular, we generalize Greene's theorem, thereby expressing harmonic weight enumerators of codes as…

Given an irreducible bivariate polynomial $f(t,x)\in \mathbb{Q}[t,x]$, what groups $H$ appear as the Galois group of $f(t_0,x)$ for infinitely many $t_0\in \mathbb{Q}$? How often does a group $H$ as above appear as the Galois group of…

Generalizing polynomials previously studied in the context of linear codes, we define weight polynomials and an enumerator for a matroid $M$. Our main result is that these polynomials are determined by Betti numbers associated with graded…

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and (b) finite and infinite matroids whose ground set have some canonical symmetry, for example row and column symmetry and transposition symmetry. For (a)…

We prove several theorems concerning Tutte polynomials $T(G,x,y)$ for recursive families of graphs. In addition to its interest in mathematics, the Tutte polynomial is equivalent to an important function in statistical physics, the Potts…

Using the theory of Newton Polygons, we formulate a simple criterion for the Galois group of a polynomial to be ``large.'' For a fixed $\alpha \in \Q - \Z_{<0}$, Filaseta and Lam have shown that the $n$th degree Generalized Laguerre…

$q$-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials $\tau(x,y)$ of matroids are calculated either by recursion over deletion/contraction of…

We study Galois descents for categories of mixed Tate motives over $\mathcal{O}_{N}[1/N]$, for $N\in \left\{2, 3, 4, 8\right\}$ or $\mathcal{O}_{N}$ for $N=6$, with $\mathcal{O}_{N}$ the ring of integers of the $N^{\text{th}}$ cyclotomic…

Let $L(x)$ be any $q$-linearized polynomial with coefficients in $\mathbb{F}_q$, of degree $q^n$. We consider the Galois group of $L(x)+tx$ over $\mathbb{F}_q(t)$, where $t$ is transcendental over $\mathbb{F}_q$. We prove that when $n$ is a…

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using…

For any matroid $M$, we compute the Tutte polynomial $T_M(x,y)$ using the mixed intersection numbers of certain classes in the combinatorial Chow ring $A^\bullet(M)$ arising from hypersimplices. Using the mixed Hodge-Riemann relations, we…

We prove new cases of the Tate conjecture for abelian varieties over finite fields, extending previous results of Dupuy--Kedlaya--Zureick-Brown, Lenstra--Zarhin, Tankeev, and Zarhin. Notably, our methods allow us to prove the Tate…

Let K be a number field and {V_l} be a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let G_l and V_l^ab be respectively the algebraic monodromy group and the maximal abelian…

Suppose \( \rho_1 \) and \( \rho_2 \) are two pure Galois representations of the absolute Galois group of a number field $K$ of weights \( k_1 \) and \( k_2 \) respectively, having equal normalized Frobenius traces \( Tr(\rho_1(\sigma_v))…

The weighted transition polynomial of a multimatroid is a generalization of the Tutte polynomial. By defining the activity of a skew class with respect to a basis in a multimatroid, we obtain an activities expansion for the weighted…