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I am stuck in the following question:

$f(x, y)$ is a bivariate polynomial with coefficients in $\mathbb Z$. 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 $f'$?2012-04-30
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    @DavidSpeyer , thanks for the hint. Can you kindly elaborate on hint 1 ? That will be helpful.2012-04-30

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