4 problems

Fix $m\geq 0$. Let $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $A_m^{(n)}$ be the set of $(n+1,m)$-excellent polynomials, meaning polynomials $f$ for which $\delta_{n+1}(f)$ is a polynom…

Filtration submodule and annihilation conjecture. For each $1\leq k<\min(m,n)$, $\mathcal{F}_{n,k}$ is an $\mathcal{A}$-submodule and a ${\mathcal{D}}_n$-submodule of…

Let $S=\mathbb{F}_q[x_1,x_2]$, let $Q=S/(x_1^{q^m},x_2^{q^m})$, and let $G=Gl_2(\mathbb{F}_q)$. In the case $q=n=2$, consider the third term, referred to in the source as the $k=3$…

Let $S=\mathbb{F}_q[x_1,\ldots,x_n]$, let $Q=S/(x_1^{q^m},\ldots,x_n^{q^m})$, and let $G=Gl_n(\mathbb{F}_q)$. Assume $m=n+1$ and consider the $k=n$ term of the conjectured Hilbert…