Let $V_n$ be the vector space of all polynomials of degree n over the complex field. I'm looking for a way to calculate the rank of the Laplacian operator $$L:\frac{\partial}{\partial x^2} + \frac{\partial}{\partial y^2}$$ for $L^i$ for $i=(1,2,3,..,r)$ where r is the number that makes the matrix nilpotent.
What I have done so far: I defined the basis (1,x,y,xy etc.) and applied the operator. I concluded that the rank will obviously be a function of n. I know that $r_n=k+1$ where $n=2k+1$ or $n=2k$. I use the fact that $Dim(ker(L^r))=dim(V_n)$. But then I'm stuck. For r-1, I though that the dimension of the kernel should be $dim(V_n) - 4$ but that isn't the case for even degrees (instead it is $dim(V_n) - 1$) and for odd it is $dim(V_n) - 3$. How do i progress from here?