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Detecting affine equivalences between certain types of parametric curves, in any dimension

Juan Gerardo Alcázar, Hüsnü Anıl Çoban, Uğur Gözütok

Abstract

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalize previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with non-rational but meromorphic components admitting a rational inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and uses bivariate factoring, instead, as a fundamental tool. The algorithm has been implemented in the computer algebra system {\tt Maple}, and can be freely downloaded and used.

Detecting affine equivalences between certain types of parametric curves, in any dimension

Abstract

Two curves are affinely equivalent if there exists an affine mapping transforming one of them onto the other. Thus, detecting affine equivalence comprises, as important particular cases, similarity, congruence and symmetry detection. In this paper we generalize previous results by the authors to provide an algorithm for computing the affine equivalences between two parametric curves of certain types, in any dimension. In more detail, the algorithm is valid for rational curves, and for parametric curves with non-rational but meromorphic components admitting a rational inverse. Unlike other algorithms already known for rational curves, the algorithm completely avoids polynomial system solving, and uses bivariate factoring, instead, as a fundamental tool. The algorithm has been implemented in the computer algebra system {\tt Maple}, and can be freely downloaded and used.
Paper Structure (14 sections, 16 theorems, 103 equations, 1 figure, 3 tables, 1 algorithm)

This paper contains 14 sections, 16 theorems, 103 equations, 1 figure, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction.
  2. Background, statement of the problem and required tools.
  3. Background, auxiliary results and statement of the problem
  4. Additional tools
  5. Development of the method, algorithm and examples
  6. Overall strategy and first step
  7. Second step (overview) and third step
  8. Algorithm and examples
  9. Justification of step (ii): computation of Möbius-commuting invariants
  10. Substep (ii.1)
  11. Substep (ii.2)
  12. Substep (ii.3)
  13. Conclusion
  14. Appendix I

Key Result

Lemma 1

Let $\Phi:{\Bbb C}^2\to {\Bbb C}^n$ be a birational mapping. Then the set of points ${\bf q}\in {\Bbb C}^2$ such that $\#(\Phi({\bf q}))>1$ is included in an algebraic variety ${\mathcal{V}}\subset{\Bbb C}^2$ of dimension at most 1.

Figures (1)

  • Figure 1: Some curves with meromorphic parametrizations, and rational inverses

Theorems & Definitions (30)

  • Lemma 1
  • proof
  • Corollary 2
  • proof
  • Example 1
  • Theorem 3
  • proof
  • Remark 1
  • Lemma 4
  • proof
  • ...and 20 more