Let $f,g\in A[x]$ with $A$ a commutative ring. Suppose $f$ is monic (for convenience), so that $A[x]/\langle f\rangle $ is a free $A$-module.
If I understand correctly, the extended Euclidean algorithm amounts to row reduction of the Sylvester matrix of $f,g$, with the bottom row of the row-reduced form containing the coordinate vector of $\gcd(f,g)$. In particular, the gcd of a monic with any polynomial always exists.
Now suppose the Sylvester matrix has nonzero kernel, i.e $uf+vg=0$ for $u,v\in A[x],\deg u<\deg g,\deg v<\deg f$. Then $uf=-vg$ is a common multiple of $f,g$ with degree $<\deg f+\deg g$. If $\operatorname{lcm}(f,g)$ exists, then its degree is also $<\deg f+\deg g$. If there's a monic gcd or lcm, the identity $\operatorname{lcm}(f,g)\gcd (f,g)=u fg,u\in A^\times$ ensures $\deg \gcd(f,g)>0$, meaning there's a common factor.
For general rings, existence of gcd does not imply existence of lcm. But I think it does hold for polynomial rings.
Question 1. Is this true?
Question 2. If $f$ is monic, when does there exist a monic gcd or lcm of $f,g$?
Question 3. In the above case, can we deduce that if $f,g\in A[x_1,\dots ,x_n][y]$ are linearly dependent with $u,v\in A[x_1,\dots ,_n][y]$ with $y$-degree bounds as above, then $\deg\gcd(f,g)>0$?
