A probabilistic algorithm to compute the real dimension of a semi-algebraic set

TL;DR

This paper introduces a probabilistic algorithm that computes the real dimension of semi-algebraic sets with single exponential complexity, significantly improving upon previous methods.

Contribution

The paper presents the first probabilistic algorithm achieving single exponential complexity for computing the real dimension of semi-algebraic sets.

Findings

Algorithm has complexity (s D)^{O(n)}

Improves over previous (s D)^{O(n^2)} complexity

Provides a positive solution to a longstanding open problem

Abstract

Let $\RR$ be a real closed field (e.g. the field of real numbers) and $\mathscr{S} \subset \RR^n$ be a semi-algebraic set defined as the set of points in $\RR^n$ satisfying a system of $s$ equalities and inequalities of multivariate polynomials in $n$ variables, of degree at most $D$, with coefficients in an ordered ring $\ZZ$ contained in $\RR$. We consider the problem of computing the {\em real dimension}, $d$, of $\mathscr{S}$. The real dimension is the first topological invariant of interest; it measures the number of degrees of freedom available to move in the set. Thus, computing the real dimension is one of the most important and fundamental problems in computational real algebraic geometry. The problem is ${\rm NP}_{\mathbb{R}}$-complete in the Blum-Shub-Smale model of computation. The current algorithms (probabilistic or deterministic) for computing the real dimension have complexity $(s \, D)^{O(d(n-d))}$, that becomes $(s \, D)^{O(n^2)}$ in the worst-case. The existence of a probabilistic or deterministic algorithm for computing the real dimension with single exponential complexity with a factor better than ${O(n^2)}$ in the exponent in the worst-case, is a longstanding open problem. We provide a positive answer to this problem by introducing a probabilistic algorithm for computing the real dimension of a semi-algebraic set with complexity $(s\, D)^{O(n)}$.