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

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 $d\ge 3$.