相关论文: Characteristic and Ehrhart polynomials

We consider the mapping $b_L\colon\mathcal{T} \to \chi$ from the Fricke-Teichm\"uller space $\mathcal{T}$ into the $\mathrm{PSL}_2\mathbb{C}$-character variety $\chi$ of the surface, obtained by bending Fuchsian representations along a…

We give an elementary proof of an analogue of Fej\'er's theorem in weighted Dirichlet spaces with superharmonic weights. This provides a simple way of seeing that polynomials are dense in such spaces.

As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side…

Hyperplane arrangements dissect $\mathbb{R}^n$ into connected components called chambers, and a well-known theorem of Zaslavsky counts chambers as a sum of nonnegative integers called Whitney numbers of the first kind. His theorem…

We introduce a weakened version of the Dunford-Pettis property, and give examples of Banach spaces with this property. In particular, we show that every closed subspace of Schreier's space $S$ enjoys it. As an application, we characterize…

Let $R$ be the polynomial ring in $n$ variables with coefficients in a field $K$ of characteristic zero. Let $D_n$ be the $n$-th Weyl algebra over $K$. Suppose that $f \in R$ defines a hyperplane arrangement in the affine space $K^n$. Then…

The Tutte polynomial is a fundamental invariant associated to a graph, matroid, vector arrangement, or hyperplane arrangement. This short survey focuses on some of the most important results on Tutte polynomials of hyperplane arrangements.…

Let $ V $ a vector space of dimension $n$. A $V$ family $ \{H_1, \ldots, H_p \} $ of vectorial hyperplanes being distinct two by two defines an arrangement $ {\cal A}_p = {\cal A} ( H_1, \ldots ,H_p ) $ of $ V $. For $ i \in \{ 1, \ldots, p…

We continue study of subgroups of a Chevalley group $G_P(\Phi,R)$ over a ring $R$ with a root system $\Phi$ and a weight lattice $P$, containing the elementary subgroup $E_P(\Phi,K)$ over a subring $K$ of $R$. Recently A. Bak and A.…

We suggest to associate with each knot the set of coefficients of its HOMFLY polynomial expansion into the Schur functions. For each braid representation of the knot these coefficients are defined unambiguously as certain combinations of…

Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic…

Weyl semimetals host relativistic chiral quasiparticles, which display quantum anomalies in the presence of external electromagnetic fields. Here, we study the manifestations of chiral anomalies in the longitudinal and planar…

We study the topological dynamics of the action of the diagonal subgroup on quotients Gamma\PSL(2,R)*PSL(2,R), where Gamma is an irreducible lattice. Closed orbits are described and a set of points of dense orbit is explicitly given. Such…

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on the extended…

Many combinatorial and topological invariants of a hyperplane arrangement can be computed in terms of its Tutte polynomial. Similarly, many invariants of a hypertoric arrangement can be computed in terms of its arithmetic Tutte polynomial.…

The lattice of intersections of reflecting hyperplanes of a complex reflection group W may be considered as the poset of 1-eigenspaces of the elements of W. In this paper we replace 1 with an arbitrary eigenvalue and study the topology and…

Consider the Wronskians of the classical Hermite polynomials $$H_{\lambda, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=\lambda_i+n-i, \,\, i=1,\dots, n$ and $\lambda=(\lambda_1,…

We present a combinatorial analysis of fiber bundles of generalized configuration spaces on connected abelian Lie groups. These bundles are akin to those of Fadell-Neuwirth for configuration spaces, and their existence is detected by a…

We recover the Tutte polynomial of a matroid, up to change of coordinates, from an Ehrhart-style polynomial counting lattice points in the Minkowski sum of its base polytope and scalings of simplices. Our polynomial has coefficients of…

If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a…