Related papers: Generalizations of Scott's inequality and Pick's f…
We discuss generalizations of some results on lattice polygons to certain piecewise linear loops which may have a self-intersection but have vertices in the lattice $\mathbb{Z}^2$. We first prove a formula on the rotation number of a…
We present an algorithm for growing the denominator $r$ polygons containing a fixed number of lattice points and enumerate such polygons containing few lattice points for small $r$. We describe the Ehrhart quasi-polynomial of a rational…
We generalize Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational simplex, that is, an n-dimensional polytope with n+1 rational vertices, we use its description as the intersection of n+1 halfspaces,…
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 prove that a rational pseudointegral triangle with exactly one lattice point in its interior has at most $9$ lattice points on its boundary, where a polygon $P$ is called pseudointegral if the Ehrhart function of $P$ is a polynomial. We…
We prove area bounds for planar convex bodies in terms of their number of interior integral points and their lattice width data. As an application, we obtain sharp area bounds for rational polygons with a fixed number of interior integral…
We give an algorithmic proof of Pick's theorem which calculates the area of a lattice-polygon in terms of the lattice-points.
In 1976, P. R. Scott characterized the Ehrhart polynomials of convex integral polygons. We study the same question for Ehrhart polynomials and quasi-polynomials of \emph{non}-integral convex polygons. Define a \emph{pseudo-integral…
Let $P$ be a polytope with rational vertices. A classical theorem of Ehrhart states that the number of lattice points in the dilations $P(n) = nP$ is a quasi-polynomial in $n$. We generalize this theorem by allowing the vertices of P(n) to…
The Ehrhart polynomial of a lattice polygon P is completely determined by the pair (b(P),i(P)) where b(P) equals the number of lattice points on the boundary and i(P) equals the number of interior lattice points. All possible pairs…
We present algorithms for classifying rational polygons with fixed denominator and number of interior lattice points. Our approach is to first describe maximal polygons and then compute all subpolygons, where we eliminate redundancy by a…
In a previous paper (El. J. Combin. 6 (1999), R37), the author generalized Ehrhart's idea of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its…
Combining $2$-descent techniques with Riemann-Roch and B\'ezout's theorems, we give an upper bound on the number of rational points of bounded height on elliptic and hyperelliptic curves over function fields of characteristic $\neq 2$. We…
Gathering different results from singularity theory, geometry and combinatorics, we show that the spectrum at infinity of a tame Laurent polynomial counts lattice points in polytopes and we deduce an effective algorithm in order to compute…
Pick's astonishing theorem explains how to obtain the area of any integer polygon by counting lattice points. It is a notoriously difficult challenge to translate the geometric statement and intuitive reasoning into a formal statement and…
This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree…
In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of…
Let $\mathcal{P} \subset \mathbb{R}^d$ be a lattice polytope of dimension $d$. Let $b(\mathcal{P})$ denote the number of lattice points belonging to the boundary of $\mathcal{P}$ and $c(\mathcal{P})$ that to the interior of $\mathcal{P}$.…
We state the formula for the critical number of vertices of a convex lattice polygon that guarantees that the polygon contains at least one point of a given sublattice and give a partial proof of the formula. We show that the proof can be…
We prove a fairly general inequality that estimates the number of lattice points in a ball of positive radius in general position in a Euclidean space. The bound is uniform over lattices induced by a matrix having a bounded operator norm.