The solutions of $\sum_{i=1}^n w_i x^{1/i} = \alpha$

Question

Let $n \in \mathbb N^* , \alpha \in \mathbb R$, and $w_i \in \mathbb R$ for all $i=1, \ldots, n$. Consider the map $f:\mathbb R_{\ge 0} \to \mathbb R$ defined by $$ f(x) := \sum_{i=1}^n w_i x^{\color{red}{1/i}}. $$

I'm interested in the solutions of $f(x)=\alpha$. I would like to search on Google Scholar, but I don't know the terminology for this kind of equation.

Could you please elaborate on the references of this kind of equation?

Answer 1

Setting $x=e^t$ we obtain an exponential sum. Exponential sums were much studied, from various points of view. One reference is Pólya–Szegő, Problems and theorems in Analysis, vol. 2, part V, Chap 1, section 6, where they are studied in the real domain. Also notice that exponential sums are solutions of linear differential equations with constant coefficients.