3 problems

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 $K$ be a field of characteristic zero, let ${\mathcal A}_K[\xi,z]$ denote the polynomial algebra in the variables $\xi$ and $z$, and let $P(\xi,z)\in {\mathcal A}_K[\xi,z]$. Le…

Let $z=(z_1,, z_n)$ be $n$ commuting variables, let $\mathbb C[z]$ be the polynomial algebra, and write $\partial=(\partial_1,\dots,\partial_n)$ with…