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Alessandro Pezzoni
Alessandro Pezzoni
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Has the following condition already been studied and, if so, is there a known class of functions that satisfy it?

Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix $$ \begin{bmatrix} 1 & x & \dotsm & x^n \\\ 1 & f & \dotsm & f^n \end{bmatrix} $$ are linearly independent over $\Bbb{Z}$, where $f: \Bbb{R} \to \Bbb{R}$ and $f \neq 0,x$.

In other words, I would like to characterise the functions $f$ for which $x^if^j - x^jf^i$, with $0 \leq i < j \leq n$, are linearly independent over $\Bbb{Z}$.

Addendum: If it simplifies the analysis, $f$ can be assumed analytic. Even a result when $f$ is a polynomial of degree bounded by $d$ would be useful.

Has the following condition already been studied and, if so, is there a known class of functions that satisfy it?

Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix $$ \begin{bmatrix} 1 & x & \dotsm & x^n \\\ 1 & f & \dotsm & f^n \end{bmatrix} $$ are linearly independent over $\Bbb{Z}$, where $f: \Bbb{R} \to \Bbb{R}$ and $f \neq 0,x$.

In other words, I would like to characterise the functions $f$ for which $x^if^j - x^jf^i$, with $0 \leq i < j \leq n$, are linearly independent over $\Bbb{Z}$.

Has the following condition already been studied and, if so, is there a known class of functions that satisfy it?

Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix $$ \begin{bmatrix} 1 & x & \dotsm & x^n \\\ 1 & f & \dotsm & f^n \end{bmatrix} $$ are linearly independent over $\Bbb{Z}$, where $f: \Bbb{R} \to \Bbb{R}$ and $f \neq 0,x$.

In other words, I would like to characterise the functions $f$ for which $x^if^j - x^jf^i$, with $0 \leq i < j \leq n$, are linearly independent over $\Bbb{Z}$.

Addendum: If it simplifies the analysis, $f$ can be assumed analytic. Even a result when $f$ is a polynomial of degree bounded by $d$ would be useful.

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Alessandro Pezzoni
Alessandro Pezzoni
  • 131
  • 3

Identifying a determinantal condition

Has the following condition already been studied and, if so, is there a known class of functions that satisfy it?

Condition. For a fixed $n > 0$, all the $2 \times 2$ minors of the matrix $$ \begin{bmatrix} 1 & x & \dotsm & x^n \\\ 1 & f & \dotsm & f^n \end{bmatrix} $$ are linearly independent over $\Bbb{Z}$, where $f: \Bbb{R} \to \Bbb{R}$ and $f \neq 0,x$.

In other words, I would like to characterise the functions $f$ for which $x^if^j - x^jf^i$, with $0 \leq i < j \leq n$, are linearly independent over $\Bbb{Z}$.