Consider two polynomials in Z[x], say $f(x)=3x^2-2x+5$ and $g(x)=2x+2$. Now consider the function $d(x)=gcd(f(x),g(x))$. So d(1) = (6,4) = 2, d(2) = (13,6) = 1, d(3) = (26,8) = 2, etc. It seems like there should be some sort of fundamental relationship between [f(x),g(x)] and d(x) beyond that described above, but I cannot for the life of me find one. I've looked at the GCD algorithm for polynomials in Z[x], thinking the mod relationship might be homomorphic, but it's easily seen very quickly that is not so. All I've been able to figure out is that if f(x) and g(x) have a polynomial divisor p(x) in common (in Z[x]), d(x) is equal p(x)*gcd(f(x)/p(x),g(x)/p(x)).
Does the world know anything else beyond this? I'm beginning to suspect the deeper connection I'm looking for doesn't exist, but all my abstract algebra is self-taught so maybe I'm missing something. Grateful to learn what the stackexchange hivemind thinks.
Regards,
--BF