Hilbert-Kamke problem

The problem of the compatibility of a system of Diophantine equations of Waring type: \begin{equation}\label{eq:1} \left.{ \begin{array}{rcl} x_1^n + \cdots + x_s^n &=& N_n \\ x_1^{n-1} + \cdots + x_s^{n-1} &=& N_{n-1} \\ \ldots&&\\ x_1 + \cdots + x_s &=& N_1 \end{array} }\right\rbrace \end{equation}

where the $x_1,\ldots,x_s$ assume integral non-negative values, certain additional restrictions [3] are imposed on the numbers $N_n,\ldots,N_1$, and $s$ is a sufficiently-large number which depends only on the natural number $4n$ which is given in advance.

The Hilbert–Kamke problem, which was posed in 1900 by D. Hilbert [1], was solved by E. Kamke, who proved that solutions to \eqref{eq:1} in fact exist. K.K. Mardzhanishvili in 1937 [3] obtained an asymptotic formula for the number of solutions of this system using the Vinogradov method for estimating trigonometric sums.

References

[1]	D. Hilbert, "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl $n^{\text{ter}}$ Potenzen (Waringsches Problem)" Math. Ann. , 67 (1909) pp. 281–300 Zbl 40.0237.02
[2]	I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) Zbl 0055.27504
[3]	K.K. Mardzhanishvili, "Sur la représentation simultanée de $n$ nombres par des sommes des puissances complètes" Izv. Akad. Nauk SSSR Ser. Mat. (1937) pp. 609–631 Zbl 63.0894.02