On complete rational trigonometric sums and integrals

Asymptotical formulae as $m\to\infty$ for the number of solutions of the congruence system of a form
$$ g_s(x_1)+\dots +g_s(x_k)\equiv g_s(x_1)+\dots +g_s(x_k)\pmod{p^m}, 1\leq s\leq n, $$
are found, where unknowns $x_1,\dots ,x_k,y_1,\dots ,y_k$ can take on values from the complete system of residues modulo $p^m,$ but degrees of polynomials $g_1(x),\dots ,g_n(x)$ do not exceed $n.$ Such polynomials $g_1(x),\dots ,g_n(x),$ for which these asymptotics hold as $2k>0,5n(n+1)+1,$ but as $2k\leq 0,5n(n+1)+1$ the given asymptotics have no place, were shew.
Besides, for polynomials $g_1(x),\dots ,g_n(x)$ with real coefficients, moreover degrees of polynomials do not exceed $n,$ the asymptotic of a mean value of trigonometrical integrals of the form
$$ \int\limits_0^1e^{2\pi if(x)}, f(x)=\alpha_1g_1(x)+\dots +\alpha_ng_n(x), $$
where the averaging is lead on all real parameters $\alpha_1,\dots ,\alpha_n,$ is found. This asymptotic holds for the power of the averaging $2k>0,5n(n+1)+1,$ but as $2k\leq 0,5n(n+1)+1$ it has no place.