A baby step-giant step roadmap algorithm for general algebraic sets

TL;DR

This paper introduces a new algorithm for constructing roadmaps of real algebraic sets with complexity bounds significantly better than previous methods, enabling more efficient analysis of their topological structure.

Contribution

It presents a novel roadmap algorithm with complexity $d^{O(k oot 2 k)}$, improving over the prior $d^{O(k^2)}$ bound, for real algebraic sets defined by polynomials.

Findings

Algorithm computes roadmaps with complexity $d^{O(k oot 2 k)}$

Efficiently determines the number of semi-algebraically connected components

Advances the computational algebraic geometry field

Abstract

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We give an algorithm that takes as input a polynomial $Q \in \mathrm{D}[X_1,\ldots,X_k]$, and computes a description of a roadmap of the set of zeros, $\mathrm{Zer}(Q,\mathrm{R}^k)$, of $Q$ in $\mathrm{R}^k$. The complexity of the algorithm, measured by the number of arithmetic operations in the ordered domain $\mathrm{D}$, is bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. As a consequence, there exist algorithms for computing the number of semi-algebraically connected components of a real algebraic set, $\mathrm{Zer}(Q,\mathrm{R}^k)$, whose complexity is also bounded by $d^{O(k \sqrt{k})}$, where $d = \mathrm{deg}(Q)\ge 2$. The best previously known algorithm for constructing a roadmap of a real algebraic subset of $\mathrm{R}^k$ defined by a polynomial of degree $d$ has complexity $d^{O(k^2)}$.