We know in a 2-D plain, $$$n$$$ point $$$(x_i,y_i)$$$ can uniquely determine a $$$n-1$$$ degree polynomial. Proof of it is to use Vandermonde Determinant.
I see a method to use 3-D Lagrange Interpolation Polynomial to determine a binary polynomial. It uses $$$(n+1)(m+1)$$$ points when the highest degree of $$$x,y$$$ is $$$n,m$$$. But why it work? I think there should be 2-D Vandermonde Determinant corresponding to it.
I find it hard to calculate the value of 2-D Vandermonde Determinant. Can Anyone help me to solve it or find a paper? Thanks :)
I have solved it similar with 1-D case.. It's $$$\prod\limits_{0\leq i<j\leq n}(x_j-x_i)^{m+1}*\prod\limits_{0\leq i<j\leq m}(y_j-y_i)^{n+1}$$$
The element is ordered lexicographically.
Can anyone solve it in higher dimensional?
It seems that in general case, each index equals to the product of other dimensional. It can be proved by induction.