# GCD of a bivariate polynomial and its partial derivative..

I am stuck in the following question :- $f(x, y)$ is a bivariate polynomial with coefficients in $Z$. We have to show that $deg(GCD(f, f_y)) > 0$ iff $deg(GCD(f, f_x)) > 0$.(Here $f_x$ denotes the partial derivative with respect to $x$.)

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Hint: Suppose that $h$ divides both $f$ and $f_x$. What can you say about the relation between $h$ and $f$? Hint 2: If $f$ is a one variable polynomial, do you know what it means when $h$ divides both $f$ and $f'$? – David Speyer Apr 30 '12 at 17:40
@DavidSpeyer , thanks for the hint. Can you kindly elaborate on hint 1 ? That will be helpful. – babu123 Apr 30 '12 at 18:00

Assume that $h=GCD(f,f_y)$. Then $f=hr$ and $hs=f_y=h_yr+hr_y$, for some polynomials $r$ and $s$. It follows that $h_yr=h(s-r_y)$.
Now, $f_x=h_xr+hr_x$. If $h,r$ are relatively prime, from $h_yr=h(s-r_y)$, it follows that $h$ divides $h_y$. But $h_y$ has smaller $y$-degree than $h$. Therefore this case is not possible. Then there is a polynomial $a$ of positive degree that divides both $h$ and $r$. This polynomial divides both $f$ and $f_x=h_xr+hr_x$.