generalizations of Vandermonde matrix to high dimensions

Question

Let $x_1,x_2,\cdots,x_n\in\mathbb{R} $ or $\mathbb{C}$. By the non-degeneracy of Vandermonde matrix

the maps

$$ f: \mathbb{R}\longrightarrow\mathbb{R}^n,$$ $$ x\longmapsto (1,x,x^2,\cdots,x^{n-1}) $$ and $$ f:\mathbb{R}^2=\mathbb{C}\longrightarrow \mathbb{R}\times\mathbb{C}^{n-1}=\mathbb{R}^{2(n-1)+1}, $$ $$ z\longmapsto (1,z,z^2,\cdots,z^{n-1}) $$ satisfy the condition:

(C) for any distinct $x_1,x_2,\cdots,x_n$, their images are linearly independent.

Question: how to construct maps satisfying (C)

$$ f: \mathbb{R}^m\longrightarrow \mathbb{R}^{m(n-1)+1} $$ for $m\geq 3$?

Answer 1

(This answers the previous version of the question.)

For example, you may choose large enough $N$, say $N=(m-1)n(n-1)/2+1$, $N$ linear functions $\ell_1,\dots,\ell_N$ in general position on $\mathbb{R}^m$ (any $m$ are linearly independent) and take the following map from $\mathbb{R}^m$ to $\mathbb{R}^{1+N(n-1)}$: $$ x\rightarrow (1,\ell_1(x),(\ell_1(x))^2,\dots,(\ell_1(x))^{n-1},\ell_2(x),\dots,(\ell_N(x))^{n-1}). $$ For any $n$ distinct points $x_1,\dots,x_n$ in $\mathbb{R}^m$ there exists index $i$ such that $\ell_i(x_1),\dots,\ell_i(x_n)$ are distinct reals. Indeed, for any pair $(x_p,x_q)$ there are at most $m-1$ indices $i$ for which $\ell_i(x_p)=\ell_i(x_q)$, totally at most $(m-1)n(n-1)/2$ not appropriate indices. Now linear independence of images follows from linear independence of $n$ Vandermonde vectors $(1,\ell_i(x_j),(\ell_i(x_j))^2,\dots,(\ell_i(x_j))^{n-1})$, $j=1,2,\dots,n$.

Answer 2

A map $f: M \to \mathbb{R}^n$ is said to be $k$-regular if whenever $x_1, \dots, x_k$ are distinct points of $M$, then $f(x_1), \dots, f(x_k)$ are independent. There is an abundance of literature on $k$-regular maps. Blagojević, Lück, and Ziegler - On highly regular embeddings gives obstructions and a nice history of the problem, as well as many references.