Suppose we have a function of the form
$$f(x)=\sum_{n=1}^{k}Q_nx^{r_n}$$
where $0<r_n<1$ and $\sum_{n=1}^kQ_n=0$ (more on actually choosing these at the end of the question). Then $f(0)=f(1)=0$. My question is, is there a way to characterize (or provide a condition for) the number of zeros this $f(x)$ has on $(0,1)$? For example, for most functions we get this type of behavior
This comes from
$$f(x)=-\frac{59 x^{917/1000}}{250}+\frac{397 x^{841/1000}}{1000}+\frac{241 x^{299/500}}{1000}-\frac{281 x^{419/1000}}{1000}-\frac{121 x^{139/500}}{1000}$$
That is, no zeros on the interval. By most, I mean I ran some tests, and got that around $70\%$ had no zeros. Around $30\%$ had one zero, and a negligible amount had two zeros. For example
which comes from
$$f(x)=\frac{59 x^{917/1000}}{250}-\frac{51 x^{841/1000}}{200}-\frac{141 x^{299/500}}{1000}+\frac{281 x^{419/1000}}{1000}-\frac{121 x^{139/500}}{1000}$$
has two zeros on the interval. I have no reason too believe that it is impossible to get even more zeros, but I have yet to find an example. Restating my question in a slightly different form, is there a way or method of choosing $Q_n$ and $r_n$ such that $f(x)$ has no zeros? Or is there a condition which guarantees this?