Computing the Volume of Compact Semi-Algebraic Sets

Let S subset of 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 >= 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.