# Proof of Chapoton's conjecture on Newton polygons of $q$-Ehrhart polynomials

## TL;DR

This paper proves Chapoton's conjecture regarding the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial for order polytopes, advancing understanding of $q$-analog combinatorics.

## Contribution

The paper provides a rigorous proof of Chapoton's conjecture on the Newton polygon shape for $q$-Ehrhart polynomials of order polytopes, a previously unconfirmed geometric property.

## Key findings

- Confirmed the shape of the Newton polygon as conjectured by Chapoton.
- Established new properties of $q$-Ehrhart polynomials related to their Newton polygons.
- Enhanced understanding of the geometric structure of $q$-analog polynomials.

## Abstract

Recently, Chapoton found a $q$-analog of Ehrhart polynomials, which are polynomials in $x$ whose coefficients are rational functions in $q$. Chapoton conjectured the shape of the Newton polygon of the numerator of the $q$-Ehrhart polynomial of an order polytope. In this paper, we prove Chapoton's conjecture.