Characterizing $f$ and $g$ such that $\deg(\gcd(f,g)) \geq 2$.
Solution 1:
Perhaps subresultants are relevant to my question.
From the wikipedia page on resultants:
...the GCD of P and Q has a degree d if and only if
${\quad\quad\displaystyle s_{0}(P,Q)=\cdots =s_{d-1}(P,Q)=0\ ,s_{d}(P,Q)\neq 0}$.
In this case, $S_d(P ,Q)$ is a GCD of $P$ and $Q$ and
$\quad\quad{\displaystyle S_{0}(P,Q)=\cdots =S_{d-1}(P,Q)=0.}$.
For example: $f=t^3-4t$ and $g=t^2+1$
Avoiding tedious hand calculations, the subresultants of $x^3-4x-\lambda$ and $x^2+1-\mu$ courtesy WA subresultants[ x^3 - 4x - \lambda , x^2 + 1 - \mu , x ]
are:
$$ \begin{align} s_0 &= λ^2 - μ^3 + 11 μ^2 - 35 μ + 25 \\ s_1 &= μ - 5 \\ s_2 &= 1 \end{align} $$
Therefore the necessary and sufficient condition for $f - \lambda$ and $g - \mu$ to have (at least) a common root is $\,s_0=λ^2 - μ^3 + 11 μ^2 - 35 μ + 25=0\,$, and for a second common root the additional condition $\,s_1=\mu-5=0\,$. The latter gives $\,\mu = 5\,$, which substituted in the former gives $\,\lambda = 0\,$.
[ EDIT ] In the simple case above, it is of course straightforward to verify the result by hand. Direct euclidean division gives $\,x^3 - 4x - \lambda = \big(x^2 + 1 - \mu\big) \cdot x + \big((\mu - 5) x - λ\big)\,$, so the condition for two common roots is $\,(\mu - 5) x - λ \equiv 0 \;\iff\; \mu-5=\lambda=0\,$.