The Computation of the Euclidean Distance Degree for the Middle Catalacticant for the Binary Forms

TL;DR

This paper computes the Euclidean distance degree of certain secant varieties to the Veronese embedding for small n, revealing new cases where EDdegree is explicitly determined for secant varieties with r ≥ 2.

Contribution

It provides the first explicit calculations of EDdegree for secant varieties to Veronese embeddings with r ≥ 2 and d ≥ 3, using the topological Aluffi-Harris formula.

Findings

EDdegree values for n=1 to 5 are 2, 7, 20, 53, 162

First explicit EDdegree computations for secant varieties with r ≥ 2 and d ≥ 3

Application of the topological Aluffi-Harris formula to these cases

Abstract

The $n$-secant varieties to the Veronese embedding $v_{2n}(\mathbb{P}^{1})$ are hypersurfaces of degree $n+1$, denoted by $\sigma_n(v_{2n}(\mathbb{P}^1))$. We compute the Euclidean distance degree $\mathrm{EDdegree}$ of $\sigma_n(v_{2n}(\mathbb{P}^1))$ for $n\le 5$ with respect to the Bombieri-Weyl quadratic form, which is maybe the most interesting case. The output for $n=1,\ldots, 5$ is respectively $2, 7, 20, 53, 162$. Our main tool is the topological Aluffi-Harris formula. This is the first case when the $\mathrm{EDdegree}$ of a $r$-secant variety to the Veronese embedding $v_{d}(\mathbb{P}^{m})$ is computed for $r\ge 2$ and $d\ge 3$.