Paper Info
Title
A probabilistic algorithm to compute the real dimension of a semi-algebraic set
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 $(s\, D)^{O(n)}$.
Year
Venue
Field
2013
CoRR
Discrete mathematics,Combinatorics,Open problem,Real closed field,Complex dimension,Ordered ring,Invariant (mathematics),Deterministic algorithm,Real algebraic geometry,Real number,Mathematics
DocType
Volume
Citations 
Journal
abs/1304.1928
0
PageRank 
References 
Authors
0.34
7
2
Name
Order
Citations
PageRank
Mohab Safey El Din145035.64
Elias P. Tsigaridas233031.01