If we have two polynomials, say $p(x)$ and $q(x)$ then we can find out their $\gcd$ by Euclidean Algorithm and also we can express the $\gcd$ as a linear combination of the polynomials, that is if $\gcd(p(x),q(x)=g(x)$ then there exists two polynomials, $a(x)$ and $b(x)$ such that $$ g(x)=p(x)a(x)+q(x)b(x) .$$ Now I want to know
Suppose that we have $m$ polynomials say, $p_1(x),p_2(x)\cdots,p_m(x)$ and their $\gcd$ is $g(x)$ then also the $\gcd$ can be expressed as the linear combination of the polynomials, but how do we express it. That is, do you have an algorithm for expressing the $\gcd$ as a linear combination of the polynomials?
Thanks.