A probabilistic algorithm to compute the real dimension of a semi-algebraic set (original) (raw)

Abstract

ABSTRACT Let RR\RRRR be a real closed field (e.g. the field of real numbers) and mathscrSsubsetRRn\mathscr{S} \subset \RR^nmathscrSsubsetRRn be a semi-algebraic set defined as the set of points in RRn\RR^nRRn satisfying a system of sss equalities and inequalities of multivariate polynomials in nnn variables, of degree at most DDD, with coefficients in an ordered ring ZZ\ZZZZ contained in RR\RRRR. We consider the problem of computing the {\em real dimension}, ddd, of mathscrS\mathscr{S}mathscrS. 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 rmNPmathbbR{\rm NP}_{\mathbb{R}}rmNPmathbbR-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))(s \, D)^{O(d(n-d))}(s,D)O(d(n−d)), that becomes (s,D)O(n2)(s \, D)^{O(n^2)}(s,D)O(n2) 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(n2){O(n^2)}O(n2) 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)(s\, D)^{O(n)}(s,D)O(n).

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