Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Hermitian Distance degree of an algebraic variety

Davide Furchì

TL;DR

This work extends Euclidean distance degree theory to the Hermitian setting by introducing the virtual Hermitian Distance degree $\mathrm{vHDdeg}$ and Hermitian Distance degree $\mathrm{HDdeg}$, which count and classify critical points of the Hermitian distance to an algebraic variety. It develops a robust algebraic framework, including the Hermitian critical ideal, Hermitian critical set $H_{\mathbf{u}}$, and the HD/ vHD correspondences, enabling both exact and bound computations for hypersurfaces and parametrized varieties. The paper provides explicit bounds (via Bernstein–Kouchnirenko and mixed volumes), a detailed study of conics, and duality relations with the dual variety $X^{\vee}$, together with a Hermitian distance polynomial $\mathrm{HDpol}_{X,\mathbf{u}}(t^2)$ encoding all distance data. It also connects the Hermitian discriminant with the complex evolute concepts, offering geometric interpretations of where the critical-point count changes. Collectively, these results illuminate the structure of nearest-point problems under Hermitian metrics and extend the toolkit for computational algebraic geometry in complex settings.

Abstract

In this paper we develop an algebraic theory to study the problem of finding the minimum distance point from an algebraic variety with respect to the Hermitian distance function. The theory generalizes the Euclidean Distance degree introduced in arXiv:1309.0049, replacing a positive symmetric bilinear form by a Hermitian form. Various examples are presented to show the robustness of the machineries.

The Hermitian Distance degree of an algebraic variety

TL;DR

This work extends Euclidean distance degree theory to the Hermitian setting by introducing the virtual Hermitian Distance degree and Hermitian Distance degree , which count and classify critical points of the Hermitian distance to an algebraic variety. It develops a robust algebraic framework, including the Hermitian critical ideal, Hermitian critical set , and the HD/ vHD correspondences, enabling both exact and bound computations for hypersurfaces and parametrized varieties. The paper provides explicit bounds (via Bernstein–Kouchnirenko and mixed volumes), a detailed study of conics, and duality relations with the dual variety , together with a Hermitian distance polynomial encoding all distance data. It also connects the Hermitian discriminant with the complex evolute concepts, offering geometric interpretations of where the critical-point count changes. Collectively, these results illuminate the structure of nearest-point problems under Hermitian metrics and extend the toolkit for computational algebraic geometry in complex settings.

Abstract

In this paper we develop an algebraic theory to study the problem of finding the minimum distance point from an algebraic variety with respect to the Hermitian distance function. The theory generalizes the Euclidean Distance degree introduced in arXiv:1309.0049, replacing a positive symmetric bilinear form by a Hermitian form. Various examples are presented to show the robustness of the machineries.
Paper Structure (13 sections, 42 theorems, 161 equations, 5 figures, 2 tables)

This paper contains 13 sections, 42 theorems, 161 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Hermitian Distance degree
  4. The Hermitian critical set
  5. vHDdeg of hypersurfaces
  6. vHDdeg of parametrized varieties
  7. HDdeg of conics
  8. HD correspondence and duality
  9. The HD discriminant
  10. Complex evolute
  11. Outward evolute
  12. The HD polynomial
  13. Conclusion

Key Result

Lemma 2.2

Let $X$ be an algebraic variety and $\mathbf{u}$ a point, then a regular critical point $\mathbf{z}\in X$ of the function $q_{\mathbf{u}}$ satisfies $\nabla_{\mathbf{z}}q_{\mathbf{u}}(\mathbf{z})\perp_{\mathbb{R}}T_{\mathbf{z}}X$.

Figures (5)

  • Figure 1: Osculating circles of the parabola $X=V(z_2-z_1^2)$ at the point $\mathbf{u}=(1/2,1/4)$ (Left) and of the ellipse $X=V(z_1^2+4z_2^2-4)$ at the point $\mathbf{u}=(0,1)$ (Right).
  • Figure 2: Outward osculating circles of the parabola $X=V(z_2-z_1^2)$ at the point $\mathbf{u}=(-1/2,1/4)$ (Left) and of the ellipse $X=V(z_1^2+4z_2^2-4)$ at the point $\mathbf{u}=(0,1)$ (Right).
  • Figure 3: Example \ref{['parab1']} - Evolute and outward evolute of $X$ (dashed line) on the real plane. The values are the number of critical points in each area.
  • Figure 4: Example \ref{['circle1']} - Evolute and outward evolute of $X$ (dashed line) on the real plane. The values are the number of critical points in each area.
  • Figure 5: Example \ref{['ellipse1']} - Evolute and outward evolute of $X$ (dashed line) on the real plane. The values are the number of critical points in each area.

Theorems & Definitions (99)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 89 more