Computing the dimension of real algebraic sets

TL;DR

This paper introduces an algorithm to compute the dimension of real algebraic sets efficiently, with practical performance often surpassing worst-case theoretical bounds, demonstrated through successful implementation on complex problems.

Contribution

The paper presents a novel algorithm for determining the dimension of real algebraic sets that leverages real geometry and topology, improving practical efficiency over existing methods.

Findings

Algorithm computes dimension with complexity depending on degrees and geometry.

Implementation successfully solves complex problems beyond current capabilities.

Practical performance often exceeds worst-case theoretical bounds.

Abstract

Let $V$ be the set of real common solutions to $F = (f_1, \ldots, f_s)$ in $\mathbb{R}[x_1, \ldots, x_n]$ and $D$ be the maximum total degree of the $f_i$'s. We design an algorithm which on input $F$ computes the dimension of $V$. Letting $L$ be the evaluation complexity of $F$ and $s=1$, it runs using $O^\sim \big (L D^{n(d+3)+1}\big )$ arithmetic operations in $\mathbb{Q}$ and at most $D^{n(d+1)}$ isolations of real roots of polynomials of degree at most $D^n$. Our algorithm depends on the real geometry of $V$; its practical behavior is more governed by the number of topology changes in the fibers of some well-chosen maps. Hence, the above worst-case bounds are rarely reached in practice, the factor $D^{nd}$ being in general much lower on practical examples. We report on an implementation showing its ability to solve problems which were out of reach of the state-of-the-art implementations.