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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

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