Computing the connected components of real algebraic curves

2026-03-17

AI Summary

This work proposes a novel algorithm rooted in real algebraic geometry and symbolic computation to address the problem of semialgebraic description of connected components of real algebraic curves. The method efficiently characterizes each connected component by constructing Boolean formulas composed of polynomial equalities and inequalities with real coefficients. Compared to the current state-of-the-art approach based on isotopy graphs, the proposed algorithm achieves a significant reduction in computational complexity, thereby offering a notable breakthrough in efficiency. This advancement enables rapid analytical processing of real algebraic structures, effectively supporting practical applications such as optical system design and robotic path planning that demand timely and precise geometric reasoning.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• connected components
• real algebraic curves
• semi-algebraic descriptions
• computational geometry
• real algebraic sets

Innovation

• real algebraic curves
• connected components
• semi-algebraic descriptions
• isotopic graph
• algorithmic complexity