Typical dimension of partial derivatives

Question

Let $V$ be the space of all homogenous polynomials over $\mathbb{C}$ in $n$ variables of degree $d$. Let $l,k$ be two integers and $f\in V$. Let $\partial^{=k}(f)$ be the space of all partial derivatives of $f$ of degree exactly $k$. I want to understand how the space of partial derivatives looks like. For example if we will allow to to multiply the partial derivatives by polynomial of degree up-to $l$ what the dimention of the space will we get? Let us define the space $$L_{k,l}(f)=span \left[ a(x)p(x): deg (a(x))=l, p(x)\in \partial^k(f)\right].$$

I wonder if it is possible to calculate the dimension of $L_{k,l}(f)$ for a typical $f$.

Answer 1

$L_{k,l}(f)$ is the $d$-th graded piece of the ideal $I$ generated by the $k$-th derivatives of $f$. Since $f$ is homogeneous, $I$ contains all the derivatives of $f$ of order $\le k$. So for general $f$ the zero locus of $I$ is empty and for $l>>0$ the space $L_{k,l}(f)$ contains all monomials of degree $k+l$.

I think that using effective versions of the Nullstellensatz one can determine explicitly $l_0$ such that the statement holds for $l\ge l_0$ , but I'm no expert.