Exact Volumes of Semi-Algebraic Convex Bodies

TL;DR

This paper introduces a method to compute the exact volumes of convex bodies defined by concave polynomial inequalities using differential equations, with an implementation in SageMath demonstrating results in low dimensions.

Contribution

A novel approach leveraging differential equations and convexity to compute volumes of semi-algebraic convex bodies with improved efficiency.

Findings

Volumes computed with arbitrary precision.

Reduction in computational steps due to convexity.

Implementation successfully applied in 2D, 3D, and 4D examples.

Abstract

We compute the volumes of convex bodies that are given by inequalities of concave polynomials. These volumes are found to arbitrary precision thanks to the representation of periods by linear differential equations. Our approach rests on work of Lairez, Mezzarobba, and Safey El Din. We present a novel method to identify the relevant critical values. Convexity allows us to reduce the required number of creative telescoping steps by an exponential factor. We provide an implementation based on the ore_algebra package in SageMath. We present examples computed with our implementation in 2, 3 and 4 dimensions.