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Euclidean distance discriminants and Morse attractors

Cezar Joiţa, Dirk Siersma, Mihai Tibăr

TL;DR

The paper investigates how the Euclidean distance discriminant behaves for complex plane curves, revealing a rich decomposition of the discriminant into focal, atypical, singular, and iflex components, and highlighting the novel phenomenon of Morse points disappearing at infinity specific to the complex setting. It develops precise definitions via the total and strict ED discriminants, and introduces attractors as limits of Morse points, then analyzes Morse numbers at infinity, at singularities, and on the regular part, deriving generic counts and focal points. The key contributions include structural theorems for $\\Delta^{\text{atyp}}$, $\\Delta^{\text{sing}}$, and $\\Delta^{\text{reg}}$, plus explicit formulas for Morse contact numbers and illustrative examples in both complex and isotropic coordinates. These results deepen the understanding of bifurcations of distance functions on complex curves and provide tools for calculating Morse attractors, with potential applications in computational algebraic geometry and related fields.

Abstract

Our study concerns the Euclidean distance function in case of complex plane curves. We decompose the ED discriminant into components which are responsible for three types of behavior of the Morse points. Besides the traditional focal component, which is non--linear; the other components are lines. In particular we shed light on the ``atypical discriminant'' which is due to the loss of Morse critical points at isotropic points at infinity. This phenomenon is specific for the complex setting. We find formulas for the number of Morse singularities which abut to the corresponding type of attractors when moving the centre of the distance function toward a point of the discriminant.

Euclidean distance discriminants and Morse attractors

TL;DR

The paper investigates how the Euclidean distance discriminant behaves for complex plane curves, revealing a rich decomposition of the discriminant into focal, atypical, singular, and iflex components, and highlighting the novel phenomenon of Morse points disappearing at infinity specific to the complex setting. It develops precise definitions via the total and strict ED discriminants, and introduces attractors as limits of Morse points, then analyzes Morse numbers at infinity, at singularities, and on the regular part, deriving generic counts and focal points. The key contributions include structural theorems for , , and , plus explicit formulas for Morse contact numbers and illustrative examples in both complex and isotropic coordinates. These results deepen the understanding of bifurcations of distance functions on complex curves and provide tools for calculating Morse attractors, with potential applications in computational algebraic geometry and related fields.

Abstract

Our study concerns the Euclidean distance function in case of complex plane curves. We decompose the ED discriminant into components which are responsible for three types of behavior of the Morse points. Besides the traditional focal component, which is non--linear; the other components are lines. In particular we shed light on the ``atypical discriminant'' which is due to the loss of Morse critical points at isotropic points at infinity. This phenomenon is specific for the complex setting. We find formulas for the number of Morse singularities which abut to the corresponding type of attractors when moving the centre of the distance function toward a point of the discriminant.

Paper Structure

This paper contains 17 sections, 14 theorems, 55 equations, 2 figures.

Table of Contents

  1. Introduction
  2. ED degree and ED discriminant
  3. Two definitions of the ED discriminant
  4. Terminology and two simple examples
  5. First step of a classification
  6. The attractors of Morse points
  7. The atypical discriminant
  8. The structure of $\Delta^{\mathop{\mathrm{atyp}}\nolimits}$
  9. Morse numbers at infinity
  10. Structure of $\Delta^{\mathop{\mathrm{sing}}\nolimits}$, and Morse numbers at the attractors of $\mathop{\mathrm{Sing}}\nolimits X$.
  11. The regular discriminant $\Delta^{\mathop{\mathrm{reg}}\nolimits}$, and Morse attractors on $X_{\mathop{\mathrm{reg}}\nolimits}$.
  12. A special role for flex points
  13. Structure of $\Delta^{\mathop{\mathrm{reg}}\nolimits}$
  14. Morse numbers at attractors on $X_{\mathop{\mathrm{reg}}\nolimits}$.
  15. Examples
  16. ...and 2 more sections

Key Result

Theorem 2.6

Let $X\subset \mathbb{C}^{2}$ be an irreducible reduced curve. Then

Figures (2)

  • Figure 1: Some real pictures of cusps, together with discriminants. The first and the third picture are genuine cups, the second has generic terms of degree 5. In all 3 cases the vertical line $\Delta_p ^{\mathop{\mathrm{sing}}\nolimits}$ is the normal at $p$. The focal sets $\Delta^{\mathop{\mathrm{focal}}\nolimits}$ behave different: the 2-3-cusp intersects $\Delta_p ^{\mathop{\mathrm{sing}}\nolimits}$ in $p$, the 2-7 cusp goes to infinity. The 2-4-cusp+5 has a finite intersection with $\Delta_p^{\mathop{\mathrm{sing}}\nolimits}$. Note that the other cusps on the focal set are not related to $p$, but to 'extrema of curvature' of $X$ outside $p$.
  • Figure 2: In blue $\Delta^{\mathop{\mathrm{reg}}\nolimits}$; in brown $\Delta^{\mathop{\mathrm{atyp}}\nolimits}$; in green a real picture of $X$.

Theorems & Definitions (24)

  • Theorem 2.6
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Theorem 3.4
  • proof
  • Corollary 3.5
  • proof
  • Theorem 3.6
  • ...and 14 more