Question 1. Take $P = \mathbb{R}[x_1,\dots, x_n] = \bigoplus_{i=0}^{\infty}V_i$, while $V_i$ are the $i$th component (vector space) of the graded ring $P$ with respect to the degree.
Define the following operator for fixed $r,i$: $f_{r}: V_i\rightarrow V_{\max{(i-r,0)}}$ such that $f_{r}(p) = (\sum_{i=1}^n\frac{\partial}{\partial^r x_i})p$
Define kernel of $f_r$ on $V_i$ as $\ker_{r,i} =\{f_r(p)=0, p\in V_i\}$.
Then what is $\dim(\ker_{r,i})$ and $\dim(\cap_{0<r} \ker_{r,i})$, or is there any explicit expression of above dimension with respect to $r,i$? The dimensions are considered to be the dimension of vector space over $\mathbb{R}$.
Question 2. Take $P = \mathbb{R}[x_1,\dots, x_n,y_1,\dots, y_n] = \bigoplus_{i=0}^{\infty}V_i$, while $V_i$ are the $i $th component (vector space) of graded ring with respect to the degree.
Define the following operator for fixed $r,s,i$: $ f_{r,s}: V_i\rightarrow V_{\max{(i-r-s,0)}}$ such that $f_{r,s}(p) = (\sum_{i=1}^n\frac{\partial}{\partial^r x_i}\frac{\partial}{\partial^s y_i})p$
Define kernel of $f_{r,s}$ on $V_i$ as $\ker_{r,s,i} =\{f_{r,s}(p)=0, p\in V_i\}$.
Then what is the number of $\dim(\ker_{r,s,i})$ and $\dim(\cap_{0<r+s}\ker_{r,s,i})$, or is ther is an explicit expression of above dimensions with respect to $r,s,i$.
Is there any reference for related topics? I think this kind of questions might have been studied. Thanks a lot.