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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?

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  • $\begingroup$ Your definition of $d'$ involves an undefined $l$. Can you look into it. $\endgroup$ Apr 28, 2015 at 0:31
  • $\begingroup$ I am guessing it should say "...degree less than $D-l$," not "...$D-d'$." $\endgroup$ Apr 28, 2015 at 16:12
  • $\begingroup$ Aren't the lowest- and the highest-degree homogeneous components of $g$ the products of those of $f$ and $g/f$? $\endgroup$ Aug 27, 2015 at 13:35

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