[Paper Review] A probabilistic algorithm to compute the real dimension of a semi-algebraic set

Mohab Safey El Din, Elias Tsigaridas|arXiv (Cornell University)|Apr 6, 2013

Polynomial and algebraic computation38 references3 citations

TL;DR

This paper presents a probabilistic algorithm that computes the real dimension of a semi-algebraic set in (sD)^O(n) arithmetic operations, achieving single exponential complexity with an exponent linear in n—resolving a longstanding open problem in computational real algebraic geometry. The method leverages quantifier elimination, limit computation via infinitesimal perturbations, and rational parametrization of critical points to efficiently determine dimension through sign condition realizability checks.

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 ${\ m NP}_{\\mathbb{R}}$-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 $(s\\, D)^{O(n)}$.

Motivation & Objective

• To address the open problem of whether the real dimension of a semi-algebraic set can be computed in single exponential time with exponent O(n) in the worst case.
• To improve upon existing algorithms with complexity (sD)^O(n²) or (sD)^O(d(n−d)), which are suboptimal when d is large.
• To provide a practical and efficient solution for computing the real dimension, a fundamental topological invariant in computational real algebraic geometry.
• To establish a framework that combines quantifier elimination, limit computation, and sign condition analysis for dimension computation.

Proposed method

• The algorithm uses a probabilistic approach based on infinitesimal perturbations and limit computation to analyze the dimension of a semi-algebraic set.
• It applies quantifier elimination over the reals to compute the limit of a parametrized semi-algebraic set as an infinitesimal parameter ε approaches zero.
• Critical points of projections are computed using rational parametrization techniques, enabling efficient limit computation and sign determination.
• The algorithm checks realizability of sign conditions via a subroutine that evaluates whether a given sign pattern corresponds to a non-empty semi-algebraic set.
• It relies on subresultant sequences for univariate polynomials over coefficient rings to perform sign computation and parametrization with controlled degree growth.
• The overall structure integrates techniques from cylindrical algebraic decomposition, roadmap construction, and infinitesimal perturbation to achieve optimal complexity.

Experimental results

Research questions

• RQ1 Can the real dimension of a semi-algebraic set be computed in single exponential time with exponent O(n) rather than O(n²)?
• RQ2 Is there a probabilistic algorithm that achieves (sD)^O(n) complexity for real dimension computation, improving upon prior bounds?
• RQ3 Can limit computation and sign condition realizability be effectively combined to determine dimension without full quantifier elimination?
• RQ4 Does the use of infinitesimal perturbations and rational parametrization enable a more efficient dimension computation than traditional methods?
• RQ5 Can the complexity of dimension computation be made output-sensitive and independent of the dimension d in the exponent?

Key findings

• The paper presents a probabilistic algorithm that computes the real dimension of a semi-algebraic set in (sD)^O(n) arithmetic operations, achieving single exponential complexity with exponent linear in n.
• This resolves the long-standing open problem of whether the real dimension can be computed in time (sD)^O(n), as posed by Koiran and Vorobjov.
• The algorithm achieves this by combining quantifier elimination over the reals with limit computation via infinitesimal perturbations and rational parametrization of critical points.
• The complexity bound is tight in terms of the number of variables n, significantly improving upon the (sD)^O(n²) bound of Koiran and the (sD)^O(d(n−d)) bound of Vorobjov.
• The method is robust and applies to real closed fields, with bit complexity bounds derived for the case Z = Z using subresultant-based arithmetic.
• The algorithm's efficiency stems from a novel use of infinitesimal parameters and limit analysis to avoid expensive full decomposition, enabling faster dimension detection.