4 problems

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…

Let $z=(z_1,z_2,\dots,z_n)$ be free commutative variables, let $D_i=\frac{\partial}{\partial z_i}$, and let … be the Laplace operator on $\mathbb C[z]$. A polynomial $P(z)$ is Hess…

Homogeneous vanishing conjecture. Both of the following vanishings should hold: 1. $\Delta^mP^{m+1}=0$ for every $m>\alpha_{[n,d]}$. 2. For every $k\geq1$, $\Delta^mP^{m+k}=0$ for…

Let $z=(z_1,\ldots,z_n)$ be commuting variables, let … be the Laplace operator, and call a formal power series $P(z)$ Hessian nilpotent if its Hessian matrix … is nilpotent. Vanish…