Multiple trigonometric sums (Q2857876)

Let \(r\geqslant 1\), \(n\geqslant 2\), \(k\geqslant 1\) and \(P\geqslant 1\) be integer numbers. Let \(J_n(k,P)\) denote the number of solutions of the system NEWLINE\[NEWLINE\begin{cases} &\displaystyle\sum_{j=1}^{2k}(-1)^jx_{1,j}^{t_1}x_{2,j}^{t_2}\dots x_{r,j}^{t_r}=0;\\ & t_1=0,1,\dots, n;\;t_2=0,1,\dots,n;\;\dots ;\;t_r=0,1,\dots,n \end{cases}NEWLINE\]NEWLINE consisting of \(M=(n+1)^r\) equations, where it is assumed that the unknown variables \(x_{i,j}\), (\(i=1,2,\dots, r\), \(j=1,2,\dots, 2k\)) belong to the set \(\{1,2,\dots, P\}\).NEWLINENEWLINEThe author proves that NEWLINE\[NEWLINE J_n(k,P)=\sigma\theta P^{2kr-\frac{rnM}{2}} +O\left(P^{2kr-\frac{rnM}{2}-\frac{1}{10M}}\right)+O\left(P^{2kr-\frac{rnM}{2}-\frac{1}{500r^2\log(rn)}}\right)NEWLINE\]NEWLINE in the case when \(k\geqslant 10Mr^2n\log(rn)\), where \(\sigma\) is some singular series and \(\theta\) is some singular integral.NEWLINENEWLINEThe result is derived using the main value theorem for multiple trigonometric sums and various suitable estimates for multiple trigonometric sums and trigonometric integrals.