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I am dealing with a generalized vandermonde matrix, and i wandered if there were simple results availiable on it's invertibility.

Fix a dimension $d$, and let $\mathbf x_1,...\mathbf x_n$ be points in $[0,1]^d$. Consider the vandermonde matrix $$M = \left\{\mathbf x_i^{\mathbf p}\right\}_{\substack{\lvert \mathbf p \rvert \le m\\ i \in 1,...n}}.$$

Where $\mathbf x^{\mathbf p} = \prod\limits_{i=1}^d x_i^{p_i}$, the $p_i$ being integers.

Fix $n,m$ so that the matrix is square, that is $n = \sum\limits_{i=0}^m \binom{i+d-1}{d-1}$.

Q : Are there conditions so that this matrix is invertible ? Maybe a ref ?

Q2 : If I simulate $\mathbf x_1,...\mathbf x_n$ uniformely from $[0,1]^d$, indepandantly, will the matrix be (almost surely) invertible ?

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  • $\begingroup$ you might want to identify the elements of $M$ more explicitly; and how should I understand $\mathbf{x}^2$ ? $\endgroup$ Jan 17, 2022 at 14:46
  • $\begingroup$ @CarloBeenakker Is it clearer like that ? $\mathbf\alpha$ wasnt very good with mathjax, $\mathbf p$ is a lot more readable. $\endgroup$
    – lrnv
    Jan 17, 2022 at 14:48
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    $\begingroup$ This is a hard question with a huge literature in numerical analysis and algebraic geometry: multivariate interpolation. For example one can generalize to the Hermite interpolation problem where we don't just impose the values of the function but also derivatives etc. A nontrivial instance is the en.wikipedia.org/wiki/Alexander%E2%80%93Hirschowitz_theorem $\endgroup$ Jan 17, 2022 at 16:06
  • $\begingroup$ @AbdelmalekAbdesselam Well this is unfortunate.. I seems from my simulation that the matrix is always invertible however. Cant we just call this matrix a change of basis in the polynomial space, in which case the question become are the polynomials $\langle \mathbf x, \mathbf X\rangle^m$ linearly independant from each other ? $\endgroup$
    – lrnv
    Jan 17, 2022 at 16:51
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    $\begingroup$ @ZachTeitler Yep i have the same for Q1 after looking at mathoverflow.net/a/192161/143783. For Q2, more general statements can be found in the ref from this answer. Thanks :) $\endgroup$
    – lrnv
    Jan 17, 2022 at 19:04

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