I'm developing theory to solve a type of polynomial which I call the 'non-standard polynomial'. I'd like to know if this is something which is already known but I'm unsure how to search for it.
Consider two $N$ dimensional vectors
$\mathbf{r}\in\mathbb{R}^N$, $\boldsymbol{\alpha}\in \mathbb{C}^N$ where elements of $\mathbf{r}$ are not necessarily integer values
then the 'polynomial', which is a function of $\theta$ is constructed as
$P(\theta) = \alpha_1^{r_1\theta} + \alpha_2^{r_2\theta}+\dots+\alpha_N^{r_N\theta}$
In my case $|\alpha_n| = 1$ for $n=1,\dots,N$.
Is anyone aware of theory around such a 'polynomial'? If so, what should I search for?
This might be called a multivariate fractional polynomial, but I doubt that there is a theory about that, for two reasons:
complex powers like $\alpha^r$ where $r$ is a real number have little meaning.
as the $\alpha_k$ and $r_k$ are unrelated and each appears once only, you in fact have a sum of complex numbers $b_k:=\alpha_k^{r_k}$.