4 papers · 1 filter
Ehrhart-Macdonald reciprocity extended
Matthias Beck, Richard Ehrenborg
For a convex polytope P with rational vertices, we count the number of integer points in integral dilates of P and its interior. The Ehrhart-Macdonald reciprocity law gives an inti…
On Stanley's reciprocity theorem for rational cones
Matthias Beck, Mike Develin
We give a short, self-contained proof of Stanley's reciprocity theorem for a rational cone K \subset R^d. Namely, let sigma_K (x) = sum_{m \in K \cap Z^d} x^m. Then sigma_K (x) and…
Coefficients and Roots of Ehrhart Polynomials
M. Beck, J. A. De Loera, M. Develin +2
The Ehrhart polynomial of a convex lattice polytope counts integer points in integral dilates of the polytope. We present new linear inequalities satisfied by the coefficients of E…
The partial-fractions method for counting solutions to integral linear systems
Matthias Beck
We present a new tool to compute the number $ϕ_\A (\b)$ of integer solutions to the linear system $$ \x \geq 0 \qquad \A \x = \b $$ where the coefficients of $\A$ and $\b$ are inte…