Secant varieties to osculating varieties of Veronese embeddings of ${\mathbb{P}}^n$

A well known theorem by Alexander-Hirschowitz states that all the higher secant varieties of $V_{n,d}$ (the $d$-uple embedding of ${\mathbb{P}}^n$) have the expected dimension, with few known exceptions. We study here the same problem for $T_{n,d}$, the tangential variety to $V_{n,d}$, and prove a conjecture, which is the analogous of Alexander-Hirschowitz theorem, for $n\leq 9$\mathrm{.}Moreover\mathrm{.}weprovethatitholdsforany$n,d$ifitholdsfor$d=3$. Then we generalize to the case of $O_{k,n,d}$, the $k$-osculating variety to $V_{n,d}$, proving, for $n=2$, a conjecture that relates the defectivity of ${\sigma }_s(O_{k,n,d})$ to the Hilbert function of certain sets of fat points in ${\mathbb{P}}^n$.