# 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 $$\Bbb{Z}$$.

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.