Let $k$ be an algebraically closed field (I do not mind to assume $k=\mathbb{C}$). Let $f(x),g(x) \in k[x]$. Hilbert's weak nullstellensatz says that exactly one of the following statements holds:
(1) $f(x)$ and $g(x)$ have a common zero $c \in k$ (namely, $f(c)=g(c)=0$).
(2) There exist polynomials $a(x),b(x) \in k[x]$ such that $af+bg=1$ (namely, $f$ and $g$ generate the unit ideal in $k[x]$).
Can one 'characterize' (= present in a general form in terms of coefficients and degrees) all $f(x),g(x)$ that do not satisfy $af+bg=1$ for all $a,b \in k[x]$, namely, $f$ and $g$ that have a common zero?
Of course, for such $f$ and $g$, the greatest common divisor is divisible by $x-d$ for some $d \in k$. But is it possible to say more? For example, if all monomials in $f$ and $g$ are of even degrees (including zero)?
Another question: What is the 'solution' to: $f,g \in k[x]$, $\deg(f),\deg(g) \geq 2$, $k(f,g)=k(x)$ (so by this question, the greatest common divisor of $f-\alpha,g-\beta$ is $x-\gamma$, for some $\alpha,\beta,\gamma \in k$) and $f',g'$ generate the unit ideal in $k[x]$: $af'+bg'=1$ for some $a,b \in k[x]$ ($f'$ is the derivative of $f$).
Thank you very much!