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Computing the connected components of real algebraic curves

Elisabetta Rocchi, Mohab Safey El Din

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.

Computing the connected components 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.
Paper Structure (14 sections, 18 theorems, 18 equations, 6 algorithms)

This paper contains 14 sections, 18 theorems, 18 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Plane Curves
  3. Outline of the construction
  4. Strategy for Assigning a Sign Pattern
  5. Algorithms and Complexity Analysis
  6. Final algorithm
  7. Space Curves
  8. Graph constructions
  9. Algorithm description and correctness
  10. Subroutines
  11. Proof of \ref{['thm:main']} (Complexity)
  12. Proof of \ref{['Twoparametrization']}
  13. Subroutines
  14. Degree, height and complexity for Lemma \ref{['lemma:P']}

Key Result

Theorem 1.1

Let ${\bm{f}}=(f_1,\dots,f_{n-1})\subset \mathbb{Q}[x_1,\dots,x_n]$ generating a radical ideal of dimension $1$, ${{\mathcal{C}}}\subset {\mathbb{C}}^n$ be the algebraic curve it defines and let $(d, h)$ be the maximum magnitude of the $f_i$'s. Let $0<\epsilon<1$ be a probability parameter. There ex

Theorems & Definitions (38)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • Corollary 2.4
  • proof
  • Lemma 2.5
  • proof : Proof of correctness
  • Proposition 2.6
  • ...and 28 more