Computing the volume of compact semi-algebraic sets - Archive ouverte HAL Access content directly
Conference Papers Year :

Computing the volume of compact semi-algebraic sets

Pierre Lairez
Mohab Safey El Din


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.
Fichier principal
Vignette du fichier
volumes.pdf (821.57 Ko) Télécharger le fichier
Origin : Files produced by the author(s)

Dates and versions

hal-02110556 , version 1 (25-04-2019)



Pierre Lairez, Marc Mezzarobba, Mohab Safey El Din. Computing the volume of compact semi-algebraic sets. ISSAC 2019 - International Symposium on Symbolic and Algebraic Computation, Jul 2019, Beijing, China. ⟨hal-02110556⟩
236 View
152 Download



Gmail Facebook Twitter LinkedIn More