Related papers: A note on palindromic $\delta$-vectors for certain…

We prove a general theorem on the existence of heteroclinic orbits in Hilbert spaces, and present a method to reduce the solutions of some P.D.E. problems to such orbits. In our first application, we give a new proof in a slightly more…

Let $F : \mathrm{End}_{\mathbb{F_p}}(\mathbb{G}_{a/K}^d)$ be an additive polynomial mapping over a global function field $K/\mathbb{F}_q$, and let $P \in \mathbb{G}_a^d(K)$. Following Silverman, consider $\delta := \lim_{n \in \mathbb{N}}…

We study the Ehrhart $h^\ast$-polynomial of (the boundary of) a lattice polytope via regular unimodular triangulations and Gr\"obner degenerations of toric ideals. Our main result is a boundary analogue of the well-known Sturmfels…

Let $P=\mathbb P^m(e)\times\mathbb P^n(h)$ be a product of weighted projective spaces, and let $\Delta_P$ be the diagonal of $P\times P$. We prove an algebraization result for formal-rational functions on certain closed subvarieties $X$ of…

We introduce the property of convex normality of rational polytopes and give a dimensionally uniform lower bound for the edge lattice lengths, guaranteeing the property. As an application, we show that if every edge of a lattice d-polytope…

Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \frac {\partial^2}{\partial z^2_i}$ the Laplace operator. A formal power series $P(z)$ is said to be {\it Hessian Nilpotent}(HN) if its Hessian matrix $\Hes P(z)=(\frac {\partial^2…

We introduce a graph structure on Euclidean polytopes. The vertices of this graph are the $d$-dimensional polytopes contained in $\mathbb{R}^d$ and its edges connect any two polytopes that can be obtained from one another by either…

Let $A \in Z^{m \times n}$, $rank(A) = n$, $b \in Z^m$, and $P$ be an $n$-dimensional polyhedron, induced by the system $A x \leq b$. It is a known fact that if $F$ is a $k$-face of $P$, then there exist at least $n-k$ linearly independent…

A simplicial polytope is combinatorially rigid if its combinatorial structure is determined by its graded Betti numbers which are important invariant coming from combinatorial commutative algebra. We find a necessary condition to be…

We show that $b$-Hurwitz numbers with a rational weight are obtained by taking an explicit limit of a Whittaker vector for the $\mathcal{W}$-algebra of type $A$. Our result is a vast generalization of several previous results that treated…

We study $A$-hypergeometric systems $H_A(\beta)$ in the sense of Gelfand, Kapranov and Zelevinsky under two aspects: the structure of their holonomically dual system, and reducibility of their rank module. We prove first that rank-jumping…

Eberhard-type theorems are statements about the realizability of a polytope (or more general polyhedral maps) given the valency of its vertices and sizes of its polygonal faces up to a linear linear degree of freedom. We present new…

Filliman duality expresses (the characteristic measure of) a convex polytope P containing the origin as an alternating sum of simplices that share supporting hyperplanes with P. The terms in the alternating sum are given by a triangulation…

It is a fundamental result in commutative algebra and invariant theory that a finitely generated graded module over a commutative finitely generated graded algebra has rational Hilbert series, and consequently the Hilbert series of the…

R. P. Stanley proved the Upper Bound Conjecture in 1975. We imitate his proof for the Ehrhart rings. We give some upper bounds for the volume of integrally closed lattice polytopes. We derive some inequalities for the delta-vector of…

We give two proofs that the $h$-vector of any paving matroid is a pure O-sequence, thus answering in the affirmative a conjecture made by R. Stanley, for this particular class of matroids. We also investigate the problem of obtaining good…

Let M be a compact, connected and simply-connected Riemannian manifold, and suppose that G is a compact, connected Lie group acting on M by isometries. The dimension of the space of orbits is called the cohomogeneity of the action. If the…

This article addresses a combinatorial problem with applications to algebraic geometry. To a convex lattice polytope $P$ and each of its integer dilations $kP$ one may associate the barycenter of its lattice points. This sequence of…

The strong spectral order induces a natural partial ordering on the manifold $H_{n}$ of monic hyperbolic polynomials of degree $n$. We prove that twisted root maps associated with linear operators acting on $H_{n}$ are G\aa rding convex on…

A unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions. It can be considered as a matrix version of exactly solvable Schr\"odinger…