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 $\,\lambda = 0\,$.