Team, Visitors, External Collaborators
Overall Objectives
Research Program
Application Domains
New Software and Platforms
New Results
Partnerships and Cooperations
XML PDF e-pub
PDF e-Pub

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 SRn 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 p0 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.