- #1
Pere Callahan
- 586
- 1
HI there,
I have a tiny question concerning the gcd of polynomials. Assume, [itex]\chi[/itex] is the greatest common divisor of the polynomails [itex]p_{ij}, i,j=1,2[/itex]. I then form
[tex]
q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd q_{22}=p_{21}^2+p_{22}^2
[/tex]
which is in fact a matrix multiplication:
[tex]
\left(\begin{array}{cc}
q_{11} & q_{12} \\ q_{21} & q_{22}
\end{array}\right) = \left(\begin{array}{cc}
p_{11} & p_{12} \\ p_{21} & p_{22}
\end{array}\right)
\left(\begin{array}{cc}
p_{11} & p_{12} \\ p_{21} & p_{22}
\end{array}\right)^T
[/tex].
I think everyone would agree that [tex]\chi^2[/itex] is a common divisor of all the q's. But is it also the greatest one...? I couldn't find a counterexample and couldn't prove it either.
Thanks a lot
I have a tiny question concerning the gcd of polynomials. Assume, [itex]\chi[/itex] is the greatest common divisor of the polynomails [itex]p_{ij}, i,j=1,2[/itex]. I then form
[tex]
q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd q_{22}=p_{21}^2+p_{22}^2
[/tex]
which is in fact a matrix multiplication:
[tex]
\left(\begin{array}{cc}
q_{11} & q_{12} \\ q_{21} & q_{22}
\end{array}\right) = \left(\begin{array}{cc}
p_{11} & p_{12} \\ p_{21} & p_{22}
\end{array}\right)
\left(\begin{array}{cc}
p_{11} & p_{12} \\ p_{21} & p_{22}
\end{array}\right)^T
[/tex].
I think everyone would agree that [tex]\chi^2[/itex] is a common divisor of all the q's. But is it also the greatest one...? I couldn't find a counterexample and couldn't prove it either.
Thanks a lot
Last edited: