The eventual vanishing criterion for Hessian-nilpotent formal power series

Let $P(z)\in {\mathbb C}[[z]]$ satisfy $o(P(z))\geq 2$, and write $\operatorname{Hes}P(z)$ for its Hessian matrix. The series $P(z)$ is Hessian nilpotent when $\operatorname{Hes}P(z)$ is nilpotent. Eventual vanishing criterion. If

$$\Delta^mP^m(z)=0$$

for all sufficiently large $m$, then $P(z)$ is Hessian nilpotent. The preceding theorem establishes analogous equivalences when the vanishing holds for every $m\geq1$, or for $1\leq m\leq n$; the eventual version is proposed as an additional criterion.