# Computing the volume of compact semi-algebraic sets

## TL;DR

This paper presents an efficient algorithm for computing the volume of compact semi-algebraic sets with arbitrary precision, leveraging periods of rational integrals and Picard-Fuchs equations, improving upon prior exponential-time methods.

## Contribution

It introduces a novel algorithm that computes volumes exactly at high precision in essentially linear time, using algebraic and differential equation techniques.

## Key findings

- Runs in essentially linear time with respect to precision p
- Improves upon previous exponential-time Monte Carlo and moment-based methods
- Assumes a conjecture for polynomial-time algebraic subroutines

## Abstract

Let $S\subset R^n$ be a compact basic semi-algebraic set defined as the real solution set of multivariate polynomial inequalities with rational coefficients. We design an algorithm which takes as input a polynomial system defining $S$ and an integer $p\geq 0$ and returns the $n$-dimensional volume of $S$ at absolute precision $2^{-p}$.Our algorithm relies on the relationship between volumes of semi-algebraic sets and periods of rational integrals. It makes use of algorithms computing the Picard-Fuchs differential equation of appropriate periods, properties of critical points, and high-precision numerical integration of differential equations.The algorithm runs in essentially linear time with respect to~$p$. This improves upon the previous exponential bounds obtained by Monte-Carlo or moment-based methods. Assuming a conjecture of Dimca, the arithmetic cost of the algebraic subroutines for computing Picard-Fuchs equations and critical points is singly exponential in $n$ and polynomial in the maximum degree of the input.