Given $m\geq 1$, a multi-degree $\alpha = (a_1, \dots, a_m) \in \mathbb{Z}_{\geq 0}^m$, and an element of $x = (x_1, \dots, x_m) \in \mathbb{C}^m$, let’s create a row of a matrix by writing down all $(a_1+1) \cdots (a_m+1)$ monomials of degrees $\leq a_i$ in the $x_i$.
As an example, for $m=2$ and $\alpha = (1,2)$, we’d get the following row given an element $x=(x_1,x_2) \in \mathbb{C}^2$:
$ ( x_1 x_2^2 \qquad x_2^2 \qquad x_1 x_2 \qquad x_2 \qquad x_1 \qquad 1). $
Choosing $A = (a_1+1) \cdots (a_m+1)$ distinct elements $x^1, \dots, x^A \in \mathbb{C}^m$, we can make a square matrix $M$ by designating a row for each $x^j = (x^j_1,\dots,x^j_m)$:
$ M := \begin{pmatrix} (x^1_1)^{a_1} \cdots (x^1_m)^{a_m} & \cdots & x^1_m & 1 \\ \vdots & \ddots & \vdots & \vdots \\ (x^A_1)^{a_1} \cdots (x^A_m)^{a_m} & \cdots & x^A_m & 1 \end{pmatrix}.$
When $m=1$, this is just the degree $a+1$ Vandermonde matrix and the determinant is the product of differences of the entries $x^j$ (in particular, it is nonzero when we choose the $x^j \in \mathbb{C}$ to be distinct). Of course, we cannot expect to have such a description in general when $m>1$.
My question is this: can we guarantee that this matrix has some minimum rank, maybe $\min\{a_i\}$ or something similar growing with degree, simply by assuming the $x^j \in \mathbb{C}^m$ are distinct? That is, can we guarantee the first $\min\{a_i\}$ rows are linearly independent, or some similar result? Some very bad bounds seem possible, depending more on $m$ than on $\alpha$, but I know there is a general theory of interpolation which should be relevant to understanding this matrix.