Greatest common divisor of polynomials

[Pere Callahan]

HI there,

I have a tiny question concerning the gcd of polynomials. Assume, $\chi$ is the greatest common divisor of the polynomails $p_{ij}, i,j=1,2$. I then form
$$q_{11}=p_{11}^2+p_{12}^2,\quadd q_{12}=p_{11}p_{21}+p_{12}p_{22},\quadd q_{21}=p_{21}p_{11}+p_{22}p_{12},\quadd q_{22}=p_{21}^2+p_{22}^2$$

which is in fact a matrix multiplication:
$$\left(\begin{array}{cc}<br /> q_{11} & q_{12} \\ q_{21} & q_{22}<br /> \end{array}\right) = \left(\begin{array}{cc}<br /> p_{11} & p_{12} \\ p_{21} & p_{22}<br /> \end{array}\right)<br /> \left(\begin{array}{cc}<br /> p_{11} & p_{12} \\ p_{21} & p_{22}<br /> \end{array}\right)^T$$.

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. <br /> <br /> Thanks a lot <img src="https://cdn.jsdelivr.net/joypixels/assets/8.0/png/unicode/64/1f642.png" class="smilie smilie--emoji" loading="lazy" width="64" height="64" alt=":smile:" title="Smile :smile:" data-smilie="1"data-shortname=":smile:" />[/tex]

 

[fresh_42]

$2 \,|\,\begin{bmatrix}26&36\\36&50\end{bmatrix} = \begin{bmatrix}1&5\\1&7\end{bmatrix} \cdot \begin{bmatrix}1&1\\5&7\end{bmatrix}=\begin{bmatrix}1&5\\1&7\end{bmatrix} \cdot \begin{bmatrix}1&5\\1&7\end{bmatrix}^\tau$