# The min degree of polynomials of two variables with a special form

Let $f(x)$, $u(x,y)$ and $v(x,y)$ be non-constant polynomials over complex number field $\mathbb{C}$. Assume that

1. $u(x,y)$ is not a polynomial only on $y$, and $v(x,y)$ is not a polynomials of only one variable;
2. $d =\deg(f(x)) \ge 5$.
3. $u(x,y)$ and $v(x,y)$ are relatively prime, and $u(x,y)$ and $f(x)$ are relatively prime.

Then is it possible that the degree of $u^d(x,y) -f(x) \cdot v^d(x,y)$ on $x$ is less than $d$?

Thank you very much for your attention.

• It sounds like you know the answer, if $d<5$. Can you share that with us? – Jyrki Lahtonen Oct 16 '14 at 9:16
• My guess would be that to answer this we need a tool telling us how well it is possible to approximate the $d$th root of $f(x)v^d$ in the ring of formal power series $\Bbb{C}[y][[x]]$ by a polynomial w.r.t. the $(x)$-adic metric. Without the approximation being accurate. Alas, I may be very wrong about this. – Jyrki Lahtonen Oct 16 '14 at 9:23