Let $f(x)=a_{k}x^{k}+\dots+a_{1}x+a_{0}\in \mathbb{R}[x]$ with $a_{k}>0$ and $N\in\mathbb{N}$ sufficiently large. I would like to know an estimate of the following sum:
$$\sum_{N\leq n\leq 2N}\exp(\alpha f(n)),$$ when $N^{-kc}\leq |\alpha|\leq N^{kC},$ where $0<c<1$ and $C>0$ (here $\exp(t)=e^{2\pi i t}$).
I know that Vinogradov's result cannot apply in this case, because we don't have control on the Diophantine approximations of $\alpha$. I tried to apply the Van der Corput's method, but the estimate is quite weak.
Does anyone know some refences on this problem?