The kernel of a Vandermonde matrix can be determined using this formula.
The following type of matrix has a similar structure, and should also have a one-dimensional kernel.
$$V= \begin{bmatrix} 1 & 1 & 1 & \ldots & 1 \\ x_1 & x_2 & x_3 & \ldots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \ldots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{m-1} & x_2^{m-1} & x_3^{m-1} & \ldots & x_n^{m-1}\\ y_1 & y_2 & y_3 & \ldots & y_n \\ y_1x_1 & y_2x_2 & y_3x_3 & \ldots & y_nx_n \\ y_1x_1^2 & y_2x_2^2 & y_3x_3^2 & \ldots & y_nx_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ y_1x_1^{m-1} & y_2x_2^{m-1} & y_3x_3^{m-1} & \ldots & y_nx_n^{m-1}\\ y_1^2x_1 & y_2^2x_2 & y_3^2x_3 & \ldots & y_n^2x_n\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ y_1^{m-1}x_1^{m-1} & y_2^{m-1}x_2^{m-1} & y_3^{m-1}x_3^{m-1} & \ldots & y_n^{m-1}x_n^{m-1}\\ \end{bmatrix} \in \mathbb{R}^{(n-1)\times n}$$
where $n = m^2+1$ and $(x_i, y_i) \neq (x_j, y_j)$ for $i \neq j$; i.e. there are $m$ groups of $m$ rows with all possible combinations of powers $y^ax^b$ and one more column than rows.
Does a similar analytical form exist for it? Or, would additional constraints be required, like $x_i^ay_i^b \neq x_i^cy_i^d$ for $i \neq j$?