Let's define, for $n,k,P\in\Bbb N$ $$ J_{k,n}(P):=\int_0^1\cdots\int_0^1\left|\sum_{x=1}^Pe(\alpha_1x+\alpha_2x^2+\cdots+\alpha_nx^n)\right|^{2k}\,d\alpha_1d\alpha_2\cdots d\alpha_n\;\;, $$ where $e(x):=e^{2\pi ix}$.
Observing now that $|z|^2=z\bar z$ and that $\int_0^1e(kx)\,dx$ equals $1$ if $k=0$ and vanishes for any other integer value of $k$, how can I show that $J_{n,k}(P)$ expresses all the integer solutions of the system $$ x_1+x_2+\cdots+x_k-x_{k+1}-x_{k+2}-\cdots-x_{2k}=0\\ x_1^2+x_2^2+\cdots+x_k^2-x_{k+1}^2-x_{k+2}^2-\cdots-x_{2k}^2=0\\ \vdots\\ x_1^n+x_2^n+\cdots+x_k^n-x_{k+1}^n-x_{k+2}^n-\cdots-x_{2k}^n=0\\ $$ where $1\le x_1,x_2,\dots,x_{2k}\le P$ ?
I'm trying to rewrite $J_{k,n}(P)$ in different ways, but I think it's something that should be seen quite quickly, not writing down all the details... but I can't see how.
Can someone helps me?
These kind of subjects are the starting point of the Vinogradov-Korobov method in the theory of the Riemann Zeta Function (I'm reading 6th chapter of Ivic Book).
The idea is to expand out the integrand; we write it as \[\prod_{j = 1}^{k} \left(\sum_{x_j = 1}^{P} e\left(\alpha_1 x_j + \cdots + \alpha_n x_j^n\right)\right) \times \prod_{j = k + 1}^{2k} \left(\sum_{x_j = 1}^{P} e\left(-\alpha_1 x_j - \cdots - \alpha_n x_j^n\right)\right).\] We then expand everything and group like terms, obtaining \[\sum_{x_1 = 1}^{P} \cdots \sum_{x_{2k} = 1}^{P} \prod_{\ell = 1}^{n} e\left(\alpha_{\ell} \left(x_1^{\ell} + \cdots + x_k^{\ell} - x_{k + 1}^{\ell} - \cdots - x_{2k}^{\ell}\right)\right).\] We now interchange the order of summation and integration. The key point is that the integral will vanish unless the coefficient of $\alpha_j$ is $0$ for all $j \in \{1,\ldots,n\}$. This happens precisely when $x_1,\ldots,x_{2k}$ are solutions to the system of equations.