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 $x$.)
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 $f_x=h_xr+hr_x$.