Discrete Mathematics & Theoretical Computer Science |

2857

- 1 Department of Mathematics
- 2 Department of Mathematics [MIT]

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 be arbitrary rational functions in $n$. In this case we prove that the number of lattice points in $P(n)$ is a quasi-polynomial for $n$ sufficiently large. Our work was motivated by a conjecture of Ehrhart on the number of solutions to parametrized linear Diophantine equations whose coefficients are polynomials in $n$, and we explain how these two problems are related.

Source: HAL:hal-01186285v1

Volume: DMTCS Proceedings vol. AN, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010)

Section: Proceedings

Published on: January 1, 2010

Imported on: January 31, 2017

Keywords: Diophantine equations,Ehrhart polynomials,lattice points,quasi-polynomials,[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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*Numerical semigroups and Kunz polytopes*

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