Constrained Triangulations, Volumes of Polytopes, and Unit Equations

TL;DR

This paper develops a combinatorial criterion for constrained triangulations of polytopes, relating their volumes to projections, and applies it to compute volumes in the context of unit equations.

Contribution

It introduces an easy-to-check criterion for constrained triangulations and establishes a volume relation via projections, with applications to unit equations.

Findings

Provides a combinatorial criterion for constrained triangulations.

Derives a volume formula relating a polytope and its shadow.

Calculates the volume of a specific polytope in unit equations theory.

Abstract

Given a polytope $\mathcal{P}$ in $\mathbb{R}^d$ and a subset $U$ of its vertices, is there a triangulation of $\mathcal{P}$ using $d$-simplices that all contain $U$? We answer this question by proving an equivalent and easy-to-check combinatorial criterion for the facets of $\mathcal{P}$. Our proof relates triangulations of $\mathcal{P}$ to triangulations of its "shadow", a projection to a lower-dimensional space determined by $U$. In particular, we obtain a formula relating the volume of $\mathcal{P}$ with the volume of its shadow. This leads to an exact formula for the volume of a polytope arising in the theory of unit equations.