Regular derivations of truncated polynomial rings

Let $\Bbbk$ be an algebraically closed field of characteristic $p>2$. Let $\mathcal{O}_n=\Bbbk[X_1,\ldots,X_n]/(X_1^p,\ldots, X_n^p)$, a truncated polynomial ring in $n$ variables, and denote by $\mathcal{L}$ the derivation algebra of $\mathcal{O}_n$. It is known that the ring of all polynomial functions on $\mathcal{L}$ invariant under the action of the group of $\mathrm{Aut}(\mathcal{L})$ is freely generated by $n$ elements. Furthermore, the related quotient morphism is faithfully flat and all its fibres are irreducible complete intersections. An element $x\in\mathcal{L}$ is called ${\it regular}$ if the centraliser of $x$ in $\mathcal{L}$ has the smallest possible dimension. In this preprint we give an explicit description of regular elements of $\mathcal{L}$ and show that a precise analogue of Kostant's differential criterion for regularity holds in $\mathcal{L}$. We also show that a fibre of the above mentioned quotient morphism is normal if and only if it consists of regular semisimple elements of $\mathcal{L}$.