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Computing the volume of compact semi-algebraic sets

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.
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https://hal.archives-ouvertes.fr/hal-02110556
Contributor : Marc Mezzarobba <>
Submitted on : Thursday, April 25, 2019 - 2:27:05 PM
Last modification on : Wednesday, March 11, 2020 - 1:13:09 AM

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  • HAL Id : hal-02110556, version 1
  • ARXIV : 1904.11705

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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⟩

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