I am trying to prove that a matrix of the following form is generically nonsingular:
$A= \begin{bmatrix} 1&1&1&1 \\ f_1 & f_2 & f_3 &f_4 \\ (f_1 -\frac{d}{dt}).f_1& (f_2 -\frac{d}{dt}).f_2 & (f_3 -\frac{d}{dt}).f_3 &(f_4 -\frac{d}{dt}).f_4 \\ (f_1 -\frac{d}{dt})^2 .f_1& (f_2 -\frac{d}{dt})^2 .f_2 & (f_3 -\frac{d}{dt})^2.f_3 &(f_4 -\frac{d}{dt})^2 .f_4 \\ \end{bmatrix}$
where $f_1 , ... , f_m $ are meromorphic or analytic functions of the undetermined variables $(x_1 , ... x_n)$. ( each $(f_i -d/dt)$ is a operator that acts from left) is there any known results on the nonsingularity of matrices of this form? note that one can write the determinant of the matrix $A$ as sum of a Vandermonde and a Wronskian determinant.
Could we prove that this sum is generically nonzero?
