Descartes rule of sign with positive real exponents
While Descartes' rule of sign works for polynomials, does it work for 'polynomials' where the exponents are real instead of natural? That is does it work for
$$5x^{\alpha} -2x^{\beta} + 3x^{\gamma} + 3 =0$$
where, assuming I have ordered them such that $\alpha>\beta>\gamma$, $\alpha, \beta, \gamma \in \mathbb{R}$?
Thank you for any help or reference.
Interesting question. Probably yes.
This makes sense only for $x>0$.
Approximate the exponents by rationals with a large denominator $n$. Then replace $x$ by $x^n$. That changes no roots and Descartes' rule of signs applies.