# Tensor valuations on lattice polytopes

## TL;DR

This paper extends classical lattice polytope valuation theories to tensor valuations, classifies low-rank equivariant tensor valuations, and identifies a rank threshold where the classification no longer applies.

## Contribution

It provides a complete classification of tensor valuations of rank up to eight on lattice polytopes, extending Ehrhart theory to tensor valuations and identifying the limits of this classification.

## Key findings

- Classified tensor valuations of rank up to eight on lattice polytopes.
- Established that all such valuations are linear combinations of Ehrhart tensors.
- Discovered that the classification breaks down at rank nine.

## Abstract

The Ehrhart polynomial and the reciprocity theorems by Ehrhart \& Macdonald are extended to tensor valuations on lattice polytopes. A complete classification is established of tensor valuations of rank up to eight that are equivariant with respect to the special linear group over the integers and translation covariant. Every such valuation is a linear combination of the Ehrhart tensors which is shown to no longer hold true for rank nine.