I am dealing with a generalized vandermonde matrix, and i wandered if there were simple results availiable on it's invertibility.
Fix a dimension $d$, and let $\mathbf x_1,...\mathbf x_n$ be points in $[0,1]^d$. Consider the vandermonde matrix $$M = \left\{\mathbf x_i^{\mathbf p}\right\}_{\substack{\lvert \mathbf p \rvert \le m\\ i \in 1,...n}}.$$
Where $\mathbf x^{\mathbf p} = \prod\limits_{i=1}^d x_i^{p_i}$, the $p_i$ being integers.
Fix $n,m$ so that the matrix is square, that is $n = \sum\limits_{i=0}^m \binom{i+d-1}{d-1}$.
Q : Are there conditions so that this matrix is invertible ? Maybe a ref ?
Q2 : If I simulate $\mathbf x_1,...\mathbf x_n$ uniformely from $[0,1]^d$, indepandantly, will the matrix be (almost surely) invertible ?