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 $\mathrm{E}\mathrm{D}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$ 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 $n=1,\dots ,5$ is respectively $2,7,20,53,162$. Our main tool is the topological Aluffi-Harris formula. This is the first case when the $\mathrm{E}\mathrm{D}\mathrm{d}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}$ of a $r$-secant variety to the Veronese embedding $v_d({\mathbb{P}}^m)$ is computed for $r\ge 2$ and $d\ge 3$.