Asymptotic formula for the mean value of a multiple trigonometric sum

When $k\ge k_0=10$ $M_{r^2}n\log(rn)$ we have for the trigonometric integral
$$ J_n(k,P)=\int_E|S(A)|^{2k}\,dA, $$
where
\begin{gather*} S(A)=\sum_{x_1=1}^P\dots\sum_{x_r=1}^P\exp(2\pi if_A(x_1,\dots,x_r)),\\ f_A(x_1,\dots,x_r)=\sum_{t_1=0}^n\dots\sum_{t_r=0}^n\alpha_{t_1\dots t_r}x_1^{t_1}\dots x_{r^r}^r \end{gather*}
and $E$ is the $M$-dimensional unit cube, the asymptotic formula
$$ J_n(k,P)=\sigma\theta P^{2kr-rnM/2}+O(P^{2kr-rnM/2-1/(2M)})+O(P^{2kr-rnM/2-1/(500r^2\log(rn))}), $$
where $\sigma$ is a singular series and $\theta$ is a singular integral.