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 $$ f:\mathbb{R}^2=\mathbb{C}\longrightarrow \mathbb{R}^{2(n-1)+1}, $$ $$ z\longmapsto (1,z,z^2,\cdots,z^{n-1}) $$ 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 $m\geq 3$?

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 $m\geq 3$?

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 $$ f:\mathbb{R}^2=\mathbb{C}\longrightarrow \mathbb{R}^{2n-1}, $$ $$ z\longmapsto (1,z,z^2,\cdots,z^{n-1}) $$ 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)

$$ f: \mathbb{R}^m\longrightarrow \mathbb{R}^{k(m,n)} $$ for $m\geq 3$? Here $k(m,n)$ is an integer depending on $m,n$.

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 $$ f:\mathbb{R}^2=\mathbb{C}\longrightarrow \mathbb{R}^{2(n-1)+1}, $$ $$ z\longmapsto (1,z,z^2,\cdots,z^{n-1}) $$ 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 $m\geq 3$?

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

the maps

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

$$ f: \mathbb{R}^m\longrightarrow \mathbb{R}^{k(m,n)} $$ for $m\geq 3$? Here $k(m,n)$ is an integer depending on $m,n$.

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 $$ f:\mathbb{R}^2=\mathbb{C}\longrightarrow \mathbb{R}^{2n-1}, $$ $$ z\longmapsto (1,z,z^2,\cdots,z^{n-1}) $$ 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)

$$ f: \mathbb{R}^m\longrightarrow \mathbb{R}^{k(m,n)} $$ for $m\geq 3$? Here $k(m,n)$ is an integer depending on $m,n$.