An estimate for exponential sums

Given a real polynomial $f(x)=a_0x^d+\cdots+a_1 x$, I want to give a sharp estimate for $\sum_{n\le X} e(f(n))$.

If $f(x)=ax$ is a linear function, we have $$\sum_{n\le X} e(\alpha n)\ll \min (X,\|\alpha\|^{-1});$$

If $f(x)\in\mathbb{Z}[x]$ where $p$ is a prime, using Weil's bound for exponential sums, we have $$\left|\sum_{n\le X} e\left(\dfrac{f(n)}{p}\right)\right|\le (d-1)\sqrt{p}.$$

For general $f(x)\in R[x]$, how large it will be? In a special cases, if $\alpha$ is a irrational number, can we get a sharp estimate for $\sum_{n\le X} e(\alpha n^2)$ or $\sum_{n\le X} e(\alpha n^d)$? I think it correlates to the rational approximate to $\alpha$. What texts can I learn?