# Is the GCD of polynomials a smooth function?

Consider the mapping $(P,Q)\mapsto \mathrm{gcd}(P,Q)$ where gcd stands for the "greatest common divisor" of two polynomials $P,Q$.

What can we say about the regularity of this function? Is it $C^1$?

As for the topology over polynomials, I just see polynomials as vectors of their coefficients.

More generally I am interested in any reference/result regarding regularity property of mapping over polynomials (but I know the zeros are continuous functions of the coefficients of the polynomials, although not $C^1$).

Consider polynomials $x-a$ and $x-b$. Then
$$\gcd(x-a,x-b)=\begin{cases}x-a & a=b\\ 1 & a \neq b\end{cases}$$