Abstract:
For $2k>0.5n(n+1)+1$$0\leq l\leq 0,5k-w-1, w=[\ln n/\ln p,]$ the asymptotic formulas was proved for the number of solutions of the system of congruences
$$
\left\{
\begin{array}{l}
x_1+\dots+x_k\equiv y_1+\dots +y_k\pmod{p^m}\ \dots\qquad\dots\qquad\dots\qquad\dots\qquad \ x_1^n+\dots+x_k^n\equiv y_1^n+\dots +y_k^n\pmod{p^m},
\end{array}
\right.
$$
where unknowns $x_1,\dots ,x_k,y_1,\dots,y_k$ run values up $1$ to $p^{m-l}$ from the complete system residues by modulo $p^{m}.$ The finding formula for $2k\leq 0.5n(n+1)+1$ has no the place.
Let be $1\leq s<r<\dots <n, s+r+\dots +n<0.5n(n+1), 0\leq l\leq 0,5k-w-1.$ Then as $2k>s+r+\dots +n$ for the number of the system of congruencies
$$
\left\{
\begin{array}{l}
x_1^s+\dots+x_k^s\equiv y_1^s+\dots +y_k^s\pmod{p^m} \ x_1^r+\dots+x_k^r\equiv y_1^r+\dots +y_k^r\pmod{p^m} \ \dots\qquad\dots\qquad\dots\qquad\dots\qquad\ x_1^n+\dots+x_k^n\equiv y_1^n+\dots +y_k^n\pmod{p^m},
\end{array}
\right.
$$
where unknowns $x_1,\dots ,x_k,y_1,\dots,y_k$ run values up $1$ to $p^{m-l}$ from the complete system residues by modulo $p^m,$ was found the asymptotic formula.
This formula has no place as $2k\leq s+r+\dots +n.$
Keywords:non-complete rational trigonometric sums, Hua Loo-keng's method, the exponent of convergence of the average value of non-complete trigonometric sums.
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