Multidimensional system of Diophantine equations (Q1676297)

The author proves an asymptotic result for the number of solutions to a system of three Diophantine equations of additive type in six variables where e ach additive summand of these equations is a simplest form whose degree in each variable does not exceed 1. \textit{L.-K. Hua} [Scientia Sin. 1, No. 1, 1--76 (1952; \url{doi:10.1360/ya1952-1-1-1})] gave an asymptotic formula for the number of solutions for \(N\to\infty\) by proving the following. Theorem 1. Let \(r'(N)\) be the number of solutions to the following system of equations: \[ \begin{cases} x_1 + x_2 + x_3 = y_1 + y_2 + y_3,\quad &x_1^2+ x_2^2+ x_3^2 \le N; \\ x_1 ^2 + x_2^2 + x_3^2 = y_1^2+ y_2^2+ y_3^2, &y_1^2+ y_2^2+ y_3^2 \le N. \end{cases} \] Then \[ r'(N) \sim \frac{35\sqrt 3}{2}N^{3/2} \ln N \quad\text{for }N\to\infty. \] V. N. Chubarikov [see \textit{G. I. Arkhipov} et al., Theory of multiple exponential sums. (Russian) Moscow: Nauka (1987; Zbl 0638.10037)] asked to find an asymptotic result for the multidimensional analogue. In this paper the author obtains the following. Theorem 2. Let \( r'(P)\) be the number of solutions to the following system of equations: \[ \begin{cases} x_1 + x_2 = x_3 + x_4, \\ y_1 + y_2 = y_3 + y_4, \\ x_1y_1 + x_2y_2 = x_3y_3 + x_4y_4, \\ 0 \le x_1, x_2, x_3, x_4, y_1, y_2, y_3, y_4 \le P. \end{cases} \] Then \[ r'(P) \sim \frac{12}{\pi^2}P^4 \ln P + O(P^4)\quad\text{for }N\to\infty. \] \{Reviewer's remark: There is an error in the first citation (at least in the translation paper): It should read ``Tarry's problem'' instead of ``Larry's problem''.\}