In a problem that I'm working on currently, the following question came up and I feel this should be fairly elementary, but I couldn't prove it myself/couldn't find a reference. Any pointers or reference to existing literature will be helpful.
Consider the exponential polynomial solution to a linear recurrence sequence (say, $u_n = \sum_{i=1}^r a_i u_{n-i}$) of order $r$ with rational coefficients, namely, $u_n = \sum_{i=1}^k p_i(n)\lambda_i^n$ where $p_i(n) = \sum_{j=1}^r p_{ij}n^j$ arise from the multiplicity of the $k$ distinct characteristic roots $\lambda_i$ of the recurrence relation.
I want to lower bound the coefficients $p_{ij}$ by solving a system of linear equations. This gives rise to the system of linear equations $Ax = b$, where $A \in \mathbb{C}^{r \times r}$ is the generalized Vandermonde matrix, $b \in \mathbb{Q}^r$, which comes from the initial conditions of the recurrence. The entries of the matrix are algebraic numbers, whose minimial polynomial is of degree at most $r$. One option is to use Cramer's rule to solve this -- it is well-known that the generalized Vandermonde matrix has a nice closed form solution, namely, $\prod_{1 \leq i < j \leq k} (\lambda_i - \lambda_j)^{m_im_j}$ where $m_i, m_j$ are respectively the multiplicities of $\lambda_i, \lambda_j$.
Does the determinant of the matrix obtained by swapping the $i$-th column of $A$ by the vector $b$ have such a nice closed form? Or at least is there a good lower bound on the absolute value of this determinant? The math.SE link above proves the invertibility of $A$. However, I would really like an exact value/lower bound for the numerator, i.e., the matrix obtained from swapping the $i$-th column with the vector $b$, as in Cramer's rule. Is there a way to give lower bounds on $p_{ij}$ without invoking Cramer's rule? The main issue here is that the known lower bounds for linear combination of algebraic numbers are quite bad (doubly exponentially small in $r$), but since there is much more structure to the algebraic numbers appearing in the matrix (the $\lambda_i$'s are conjugates of a degree $r$ polynomial), I am hoping there is a better lower bound.
