Let $k$ be a field. Given a form $f \in k[x_1, ..., x_n]$ of degree at least $2$, we define the Schmidt rank, also known as the $h$-invariant, $h_k(f)$ to be the least positive integer $h$ such that $f$ can be written identically as $$ f = u_1 v_1 + ... + u_h v_h, $$ where each $u_i$ and $v_i$ are forms in $k[x_1, ..., x_n]$ of degree at least $1$. For example, this comes up in the paper "The density of integer points on homogeneous varieties" by Wolfgang Schmidt.
It can be easily shown that $$ h-1 \leq h_k(f |_{x_j = 0}) \leq h \leq n. $$ Let $i_1, ..., i_n$ be the integers $1$ to $n$ in any order. We have by the above inequalities that $$ h = h_k({f}) \geq h_k({f} |_{x_{i_1}=0}) \geq h_k( {f} |_{x_{i_1}, x_{i_2}=0}) \geq ... \geq h_k( {f} |_{x_{i_1}, x_{i_2}, ..., x_{i_n} =0}) = 0, $$ where at each step in the above chain of inequalities, either the value stays the same or decreases by $1$.
I am interested in proving the following: Let $f \in k[x_1, ..., x_n]$ be a form of degree $d>1$. Suppose positive integers $C$ and $M$ are given, and $h = h_k(f) > XC$ (where $X$ is sufficiently large with respect to $M$). Suppose also that $h_k(\mathbf{f}|_{x_j = 0}) = h - 1$ for all $1 \leq j \leq n$. Then there exists $M' = M'(X,M,f)>0$ and $\mathbf{x}' = (x_{i_1}, ..., x_{i_{MC}})$ such that $$ h_k(\mathbf{f}|_{\mathbf{x}'= \mathbf{0} }) \leq h - M' C. $$
I am guessing that the statement is true, and I am very interested in proving it but I don't even know where to start at the moment... I would appreciate any assistance! I would also appreciate any input or reference! Thank you very much!
PS I also tagged commutative algebra and algebraic geometry in case there might be a more algebraic/geometric approach to this. Thanks!
