I would like to know what the current best estimation for the upper bound of the exponential sum $$\left|\sum_{n=1}^N \exp \left(2 \pi i\alpha\left(x_0+x_1 n+\ldots+x_d n^d\right)\right)\right|=\left|\sum_{n=1}^N e(P(n)\alpha)\right|$$ that only require $\alpha$ to be an irrational number and fix a given $d\geq 2$ and a polynomial with integer coefficients $P(n)=x_0+x_1 n+\ldots+x_d n^d$, where $x_i\in \mathbb{N}$ and $x_d\neq0$, and I have not found any particularly good references.

From the work of Weyl and Van der Corput, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{2^d}$

From the work of Vinogrodov, we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{C d^2 \log d}$

From the work of Decoupling (J Bourgain, C Demeter, L Guth), we know

$\left|\sum_{n=1}^N e(P(n)\alpha)\right| \lesssim_{d, \epsilon, \alpha} N^{1+\epsilon}\left(q^{-1}+N^{-1}+q N^{-k}\right)^\frac{1}{d(d-1)}$

Thank you in advance.