Related papers: Spanning Lattice Polytopes and the Uniform Positio…

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in…

We completely classify non-spanning $3$-polytopes, by which we mean lattice $3$-polytopes whose lattice points do not affinely span the lattice. We show that, except for six small polytopes (all having between five and eight lattice…

Ehrhart discovered that the function that counts the number of lattice points in dilations of an integral polytope is a polynomial. We call the coefficients of this polynomial Ehrhart coefficients, and say a polytope is Ehrhart positive if…

Reflexive lattice polytopes play a key role in combinatorics, algebraic geometry, physics, and other areas. One important class of lattice polytopes are lattice simplices defining weighted projective spaces. We investigate the question of…

In our earlier paper [9], it is proved that a homogeneous rigid, traction or impedance condition on one or two intersecting line segments together with a certain zero point-value condition implies that the solution to the Lam\'e system must…

The Lieb-Schultz-Mattis theorem dictates that a trivial symmetric insulator in lattice models is prohibited if lattice translation symmetry and $U(1)$ charge conservation are both preserved. In this paper, we generalize the…

The special linear group G=SL_n(Z[x1,...,xk]) (n at least 3 and k finite) is called the universal lattice. Let n be at least 4, p be any real number in (1,\infty). The main result is the following: any finite index subgroup of G has the…

We consider $d$-dimensional lattice polytopes $\Delta$ with $h^*$-polynomial $h^*_\Delta=1+h_k^*t^k$ for $1<k<(d+1)/2$ and relate them to some abelian subgroups of $\SL_{d+1}(\C)$ of order $1+h_k^*=p^r$ where $p$ is a prime number. These…

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of…

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…

This paper is to study the Ehrhart function $L(P,t)$ of a rational $n$-polytope $P$, defined as the number of lattice points of dilated polytopes $tP$ with real numbers $t\geq 0$. It turns out that $L(P,t)$ is a quasi-polynomial of real…

We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices when all agents have path-independent choice functions. This result answers an open question of Blair~\cite{blair1988lattice}. In the…

We consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary. We estimate the $L^{p}$ norm of…

Let $\Delta$ be an $n$-dimensional lattice polytope. The smallest non-negative integer $i$ such that $k \Delta$ contains no interior lattice points for $1 \leq k \leq n - i$ we call the degree of $\Delta$. We consider lattice polytopes of…

Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness…

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…

In this paper, we show the existence of universal inequalities for the $h^*$-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the $h^*$-polynomial which are independent of both the…

The presence of flat bands is a source of localization in lattice systems. While flat bands are often unstable with respect to interactions between the particles, they can persist in certain cases. We consider a diamond ladder with…

We prove a sharp upper bound on the number of boundary lattice points of a rational polygon in terms of its denominator and the number of interior lattice points, generalizing Scott's inequality. We then give sharp lower and upper bounds on…

In this article we compare the set of integer points in the homothetic copy $n\Pi$ of a lattice polytope $\Pi\subseteq\R^d$ with the set of all sums $x_1+\cdots+x_n$ with $x_1,...,x_n\in \Pi\cap\Z^d$ and $n\in\N$. We give conditions on the…