Related papers: Obstructions to combinatorial formulas for plethys…
A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…
We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme,…
By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R^d. In this paper we study the following question: does the density hold if we approximate only by…
We compute the special values of partial zeta function at $s=0$ for family of real quadratic fields $K_n$ and ray class ideals $\fb_n$ such that $\fb_n^{-1} = [1,\delta(n)]$ where the continued fraction expansion of $\delta(n)$ is purely…
We consider the plethysm problem stated for representations of symmetric groups. In particular, we prove new relationships between composition multiplicities of twisted Foulkes modules. Expressed in terms of symmetric functions, our results…
In this paper, we first consider the graph of $(F_1,F_{2},\cdots,F_{n})$ on $\overline{\mathbb{D}}^{n},$ where $F_{j}(z)=\bar{z}^{m_{j}}_{j}+R_{j}(z),j=1,2,\cdots,n,$ which has non-isolated CR-singularities if $m_{j}>1$ for some…
It has been known since 2007 that the Wilson and Racah polynomials can be characterized as basis functions for irreducible representations of the quadratic symmetry algebra of the quantum superintegrable system on the 2-sphere,…
The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated…
We prove that given a closed connected symplectic manifold equipped with a Borel probability measure, an arbitrarily large portion of the measure can be covered by a symplectically embedded polydisk, generalizing a result of Schlenk. We…
In an earlier paper, we used the absolute grading on Heegaard Floer homology to give restrictions on knots in $S^3$ which admit lens space surgeries. The aim of the present article is to exhibit stronger restrictions on such knots, arising…
The univariate Ehrhart and $h^*$-polynomials of lattice polytopes have been widely studied. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and $h^*$-polynomials of lattice polytropes, which…
Multivariate piecewise polynomial functions (or splines) on polyhedral complexes have been extensively studied over the past decades and find applications in diverse areas of applied mathematics including numerical analysis, approximation…
We extend to infinite dimensional Hilbert spaces a celebrated result, due to B. Polyak, about the convexity of the joint image of quadratic functions. We give sufficient conditions which assure that the joint image is also closed. However,…
We establish the Plancherel-Rotach-type asymptotics around the largest zero (the soft edge asymptotics) for some classes of polynomials satisfying three-term recurrence relations with exponentially increasing coefficients. As special cases,…
We answer some questions related to multiplicity formulas by Rosenthal and Zelevinsky and by Lakshmibai and Weyman for points on Schubert varieties in Grassmannians. In particular, we give combinatorial interpretations in terms of…
Given an integral $d \times n$ matrix $A$, the well-studied affine semigroup $\mbox{ Sg} (A)=\{ b : Ax=b, \ x \in {\mathbb Z}^n, x \geq 0\}$ can be stratified by the number of lattice points inside the parametric polyhedra $P_A(b)=\{x:…
We study the class of (locally) anti-blocking bodies as well as some associated classes of convex bodies. For these bodies, we prove geometric inequalities regarding volumes and mixed volumes, including Godberson's conjecture, near-optimal…
The number of lattice points $\left| tP \cap \mathbb{Z}^d \right|$, as a function of the real variable $t>1$ is studied, where $P \subset \mathbb{R}^d$ belongs to a special class of algebraic cross-polytopes and simplices. It is shown that…
Generalizing the notion of Newton polytope, we define the Newton-Okounkov body, respectively, for semigroups of integral points, graded algebras, and linear series on varieties. We prove that any semigroup in the lattice Z^n is…
This paper is devoted to the study of general (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial…