3 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…

Symmetry conjecture for Laplacian neighbours. The relation $\sim_L$ is symmetric: whenever $z\sim_L x$, one also has $x\sim_L z$.

Let $D\subset\mathbb{R}^n$ be an optimization domain for the Neumann problem for the Laplace operator, and let $\varphi$ be a scalar phase-field with interpolation coefficient $a$…