I have a matrix which is similar to Vandermonde matrix except that the entries are monomials of degree $d$ polynomial in 2 variables. Each row has the following form:

$X_{i}= [1, x_{i}, y_{i}, x_{i}^2, x_{i}y_{i}, y_{i}^2, x_{i}^3, x_{i}^2y_{i},...,x_{i}y_{i}^{d-1}, y_{i}^{d}]$

so that the matrix $X$ contains $n$ rows, $X_{1},...,X_{n}$ where $n$ is large enough that I have more rows than columns.

My question is: when does the matrix $X$ have full rank?

What I'm looking for is necessary/sufficient conditions for when the columns are linearly independent. If this were a Vandermonde matrix (drop all columns that include a $y$ term for example), then the columns would be independent provided that I had enough distinct $x$ values (where enough is the number of columns in the Vandermonde matrix -1). Is there a similar condition here? How many distinct $x$ and $y$ values might guarantee such a result?

For information, the matrix arises in polynomial linear regression. We have a bunch of data points $(x_{i}, y_{i}, z_{i})$ and we want to build a best-fit polynomial surface. The coefficients of the polynomial come from $(X^T X)^{-1}X^T Z$ and so we want to be certain that $X^T X$ is invertible.