# Are fractional polynomials linearly independent?

1. Apr 21, 2014

### dipole

i.e., does the set of functions of the form,

$\{ x^{\frac{n}{m}}\}_{n=0}^{\infty}$ for some fixed $m$ produce a linearly independent set? Either way, can you give a brief argument why or why not?

Just curious :)

2. Apr 21, 2014

### disregardthat

If you have any linear combination between a finite number of these elements, this relation can be written as

$$a_ny^n+a_{n-1}y^{n-1}+...+a_1y = 0$$

where $y = x^{\frac{1}{m}}$ and a_n is non-zero. However, a polynomial of degree n has exactly n zeroes, which means that this is impossible.