4
$\begingroup$

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 $C^1$).

$\endgroup$
5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.