Consider two polynomials P and Q and their companion matrices. It seems that char polynomial of tensor product of said matrices would be a polynomial with roots that are all possible pairs product of roots of P,Q.

I guess its coefficients could be expressed through coefficients of P and Q. But I don't know the explicit formula and I cannot find it. I also failed to find it out myself -- I tried different approaches. Maybe it should be that characteristic polynomial, maybe resultant of some form, but..

I hope this is done by someone already.

I'll try to give an example $c_0 = a_0^nb_0^m$, or maybe $c_0 = (-1)^{n+m}a_0^nb_0^m$ ($m,n$ are the degrees of $P and Q$.

$c_1 = a_0^{n-1}a_1b_0^{m-1}b_1$, this is one is probably incorrect, but I obviously cannot provide the correct answer if I don't know it yet. It looks like $c_i$ is a sum of some sort.