Section: New Results

Computing the Volume of Compact Semi-Algebraic Sets

In [10], Pierre Lairez, Mohab Safey El Din and Marc Mezzarobba join a unique set of expertise in symbolic integration, real algebraic geometry and numerical integration to tackle a problem as old as Babylonian mathematics: the computation of volumes.

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. They design an algorithm which takes as input a polynomial system defining $S$ and an integer $p\ge 0$ and returns the $n$-dimensional volume of $S$ at absolute precision $2^{-p}$.

Their 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.