Let $f(x,y,z) = ax^d + b y^k z^{d-k} + c$, where $a,b,c \in \mathbb{C}$ and $abc \ne 0$, and $d \geq 3$. Let $d'$ denote the smallest non-negative integer $l$ such that there does not exist a positive integer $D$ and a polynomial $g(x,y,z) \in \mathbb{C}[x,y,z]$ of degree $D$ with the property that $g$ does not have any monomials of degree less than $D - d'$, and $f(x,y,z) | g(x,y,z)$.
Clearly, $d' \geq 0$, since if $d' = 0$ then $g$ is forced to be homogeneous, and thus cannot have a factor which is not homogeneous.
Is the estimate $d' = 0$ sharp in general? Or can this be improved?
