Let $p_1, p_2,\dots, p_n$ and $q_1,q_2,\dots,q_n$ be a collection of complex polynomials. Let $A$ be a $n \times n$ matrix satisfying

$$a_{ij} = \begin{cases} p_i(x) & \text{ if } i = j, \\ q_i(x) & \text{ otherwise} \end{cases} .$$

is there any connection between the roots of the polynomials $p_i$'s and $q_i$'s and the roots of the polynomial $\det A$? if not, is this true under at least under any special assumptions?

Kindly share some references.

Thank you.