happyguy656's blog

By happyguy656, history, 3 years ago, In English

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 :)

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3 years ago, # |
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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?

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3 years ago, # |
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It seems that in general case, each index equals to the product of other dimensional. It can be proved by induction.