Computing the connected components of real algebraic curves

TL;DR

This paper introduces a new, more efficient algorithm for computing semi-algebraic descriptions of real algebraic curves, improving upon existing methods in terms of complexity.

Contribution

The paper presents a novel algorithm that reduces the complexity of computing semi-algebraic descriptions of real algebraic curves compared to previous approaches.

Findings

Algorithm has lower complexity than previous methods

Enables applications in optical system design and robotics

Provides explicit semi-algebraic descriptions of real algebraic curves

Abstract

Connected components of real algebraic sets are semi-algebraic, i.e. they are described by a boolean formula whose atoms are polynomial constraints with real coefficients. Computing such descriptions finds topical applications in optical system design and robotics. In this paper, we design a new algorithm for computing such semi-algebraic descriptions for real algebraic curves. Notably, its complexity is less than the best known one for computing a graph which is isotopic to the real space curve under study.