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

$$\gcd(x-a,x-b)=\begin{cases}x-a & a=b\\ 1 & a \neq b\end{cases}$$

This is definitelly not continuous in your topology.

  • $\begingroup$ You're right, my question was silly... Thx! $\endgroup$ – Student Jun 3 '17 at 15:22
  • 3
    $\begingroup$ @Student Questions cannot be silly. The only silly thing is not asking questions. Glad to be of help! $\endgroup$ – lisyarus Jun 3 '17 at 15:28

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