Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Osculating Geometry and Higher-Order Distance Loci

Sandra Di Rocco, Kemal Rose, Luca Sodomaco

Abstract

We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order tangency. We focus on the data locus of points possessing at least one critical point of the distance function that is normal to a higher-order osculating space. We study the higher-order distance degree of a morphism as an intersection-theoretic invariant involving jet bundles and higher-order polar classes. Our approach builds on foundational definitions and results developed by Piene, particularly regarding higher-order polar loci. We give closed formulas for generic maps, Veronese embeddings, and toric embeddings. We place particular emphasis on the Bombieri-Weyl metric, revealing that the chosen metric profoundly influences both the degree and birationality of the higher-order projection maps. Additionally, we introduce a tropical framework that represents these degrees as stable intersections with Bergman fans, facilitating effective combinatorial computation in toric settings.

Osculating Geometry and Higher-Order Distance Loci

Abstract

We discuss the problem of optimizing the distance function from a given point, subject to polynomial constraints. A key algebraic invariant that governs its complexity is the Euclidean distance degree, which pertains to first-order tangency. We focus on the data locus of points possessing at least one critical point of the distance function that is normal to a higher-order osculating space. We study the higher-order distance degree of a morphism as an intersection-theoretic invariant involving jet bundles and higher-order polar classes. Our approach builds on foundational definitions and results developed by Piene, particularly regarding higher-order polar loci. We give closed formulas for generic maps, Veronese embeddings, and toric embeddings. We place particular emphasis on the Bombieri-Weyl metric, revealing that the chosen metric profoundly influences both the degree and birationality of the higher-order projection maps. Additionally, we introduce a tropical framework that represents these degrees as stable intersections with Bergman fans, facilitating effective combinatorial computation in toric settings.

Paper Structure

This paper contains 13 sections, 31 theorems, 186 equations, 8 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Osculating functions
  3. A gallery of toric examples
  4. Higher-order polar classes
  5. Higher-order distance loci
  6. Osculating eigenvectors of symmetric tensors
  7. Proof of Theorem \ref{['thm: projection data locus Veronese BW is either birational or of degree 2']}
  8. Proof of Theorem \ref{['thm: generic ED degree general polynomial map']}
  9. Higher-order distance degrees of regular embeddings
  10. Tropical geometry of distance optimization
  11. Proof of Theorem \ref{['theorem: tropical description of higher-order multidegrees']}
  12. Toric varieties
  13. Higher-order distance degrees of affine morphisms

Key Result

Theorem 1.1

Let $f_0,\dots,f_n$ be $n+1$ generic polynomials in $\mathbb{C}[x_0,\dots,x_m]_d$ and let $f\colon\mathbb{P}^m\to\mathbb{P}^n$ be the associated morphism. Consider a nonnegative integer $k\le d$ and assume that $n>\binom{m+k}{k}-1$. Then $f$ has generic $k$-osculating dimension $\binom{m+k}{k}-1$. F

Figures (8)

  • Figure 1: Tangent lines and osculating planes of the twisted cubic image of the Veronese embedding $f=\nu_1^3\colon\mathbb{P}^1\hookrightarrow\mathbb{P}^3$ in the affine chart $\{u_0=1\}\cong\mathbb{R}^3$.
  • Figure 2: Second-order distance loci of the Veronese embedding $f=\nu_1^3\colon\mathbb{P}^1\hookrightarrow\mathbb{P}^3$, with respect to the standard (left) and Bombieri-Weyl (right) quadratic form in $S^3\mathbb{R}^2$, displayed in the real affine chart $\{u_0=1\}$.
  • Figure 3: The polygon and the vectors of exponents of the monomial map representing the embedding $f_1$ of Example \ref{['ex: blowup P2 three points']}.
  • Figure 4: The polygon and the vectors of exponents of the monomial maps representing the embedding $f_2$ of Example \ref{['ex: blowup P1xP1 one point']}.
  • Figure 5: The projective surfaces $\operatorname{DL}_2(f',Q_\mathsmaller{\mathrm{ED}}')\cap\mathbb{P}(V)$ and $\operatorname{DL}_2(f, Q_\mathsmaller{\mathrm{ED}})$ in the affine chart $\{u_3=1\}\subseteq\mathbb{P}^3$.
  • ...and 3 more figures

Theorems & Definitions (100)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 90 more