Related papers: Highest coefficients of weighted Ehrhart quasi-pol…
We give a $K$-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope $P$, depending on a given homogeneous polynomial function $\varphi$ on $P$, and with Laurent polynomial…
We describe a method to evaluate multivariate polynomials over a finite field and discuss its multiplicative complexity.
We give an Euler-Maclaurin formula with remainder for the weighted sum of the values of a smooth function on the integral points in a simple integral polytope. Our work generalizes the formula obtained by Karshon, Sternberg and Weitsman in…
The Cauchy polynomials with a $q$ parameter were recently defined, and several arithmetical properties were studied. In this paper, we establish explicit formulae for computing the Cauchy polynomials with a $q$ parameter in terms of…
We compute the spectra of the adjacency matrices of the semi-regular polytopes. A few different techniques are employed: the most sophisticated, which relates the 1-skeleton of the polytope to a Cayley graph, is based on methods akin to…
In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led us to define certain polynomials analogous to Ehrhart…
The Al-Salam-Chihara polynomials are an important family of orthogonal polynomials in one variable $x$ depending on 3 parameters $\alpha$, $\beta$ and $q$. They are closely connected to a model from statistical mechanics called the…
We present an algorithm for computing asymptotic approximations of roots of polynomials with exp-log function coefficients. The real and imaginary parts of the approximations are given as explicit exp-log expressions. We provide a method…
Orthogonal polynomials in two variables on cubic curves are considered, including the case of elliptic curves. For an integral with respect to an appropriate weight function defined on a cubic curve, an explicit basis of orthogonal…
We consider Hilbert-type functions associated with difference (not necessarily inversive) field extensions and systems of algebraic difference equations in the case when the translations are assigned some integer weights. We will show that…
Laguerre and Laguerre-type polynomials are orthogonal polynomials on the interval $[0,\infty)$ with respect to a weight function of the form $w(x) = x^{\alpha} e^{-Q(x)}, Q(x) = \sum_{k=0}^m q_k x^k, \alpha > -1, q_m > 0$. The classical…
We use a version of localization in equivariant cohomology for the norm-square of the moment map, described by Paradan, to give several weighted decompositions for simple polytopes. As an application, we study Euler-Maclaurin formulas.
It is already known that order polytopes and chain polytopes are always 2-level polytopes. In general, this is not true for marked order and marked chain polytopes. We study the geometry of marked order polytopes, marked chain polytopes,…
We present a bounded probability algorithm for the computation of the Chow forms of the equidimensional components of an algebraic variety. Its complexity is polynomial in the length and in the geometric degree of the input equation system…
The Ehrhart quasi-polynomial of a rational polytope $P$ is a fundamental invariant counting lattice points in integer dilates of $P$. The quasi-period of this quasi-polynomial divides the denominator of $P$ but is not always equal to it:…
A positroid is a matroid realized by a matrix such that all maximal minors are non-negative. Positroid polytopes are matroid polytopes of positroids. In particular, they are lattice polytopes. The Ehrhart polynomial of a lattice polytope…
One considers weighted sums over points of lattice polytopes, where the weight of a point v is the monomial q^f(v) for some linear form f. One proposes a q-analogue of the classical theory of Ehrhart series and Ehrhart polynomials,…
We solve the connection coefficient problem between the Al-Salam-Chihara polynomials and the q-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some…
We study the maximum weight convex polytope problem, in which the goal is to find a convex polytope maximizing the total weight of enclosed points. Prior to this work, the only known result for this problem was an $O(n^3)$ algorithm for the…
The paper presents linear integral predictors for continuous time high-frequency signals with a a finite spectrum gap. The predictors are based on approximation of a complex valued periodic exponential (complex sinusoid) by rational…