For $1 \le i, j \le k$, consider $\rho_{ij}$ which are equal to either zero or one such that $\rho_{ii}=1$ and $\rho_{ij}=0$ if and only if $\rho_{ji}=0$. How to find the zeros of the determinant of the following matrix?
$$\begin{bmatrix} g_1(x) & -\rho_{12}f_1(x) & \cdots & -\rho_{1,k}f_1(x) \\ -\rho_{21}f_2(x) & g_2(x) & \cdots & -\rho_{2,k}f_2(x) \\ \vdots & \vdots & \ddots & \vdots \\ -\rho_{k1}f_k(x) & -\rho_{k2}f_k(x) & \cdots &g_k(x) \end{bmatrix}$$
where $f_i$s are $g_i$s are complex polynomials. When all the $\rho_{ij}=1$, I have got the answer in MO already at Roots of determinant of matrix with polynomial entries. The present question is a more general case.
I am not sure whether there is any direct answer to this question. I am looking for some reference where similar kinds of problems ars discussed.
