Osculating Geometry and Higher-Order Distance Loci

TL;DR

This paper explores the algebraic and geometric properties of higher-order distance loci, focusing on their degrees, invariants, and computational methods, especially under the Bombieri-Weyl metric and tropical geometry frameworks.

Contribution

It introduces formulas for higher-order distance degrees, analyzes the impact of metrics on these degrees, and develops a tropical approach for combinatorial computation.

Findings

Closed formulas for generic, Veronese, and toric maps

The Bombieri-Weyl metric affects degree and birationality

A tropical framework enables combinatorial calculations

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 loci. 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.