Difference between revisions of "Hilbert-Kamke problem"
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| − | The problem of the compatibility of a system of [[ | + | The problem of the compatibility of a system of [[Diophantine equations]] of [[Waring problem|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 [[#References|[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 [[#References|[1]]], was solved by E. Kamke, who proved that solutions to \eqref{eq:1} in fact exist. K.K. Mardzhanishvili in 1937 [[#References|[3]]] obtained an asymptotic formula for the number of solutions of this system using the [[Vinogradov method]] for estimating trigonometric sums. | |
| − | + | ====References==== | |
| + | <table> | ||
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> 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}}</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian) {{ZBL|0055.27504}}</TD></TR> | ||
| + | <TR><TD valign="top">[3]</TD> <TD valign="top"> 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}}</TD></TR> | ||
| + | </table> | ||
| − | + | {{TEX|done}} | |
| − | |||
Latest revision as of 19:09, 10 October 2017
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 |
How to Cite This Entry:
Hilbert-Kamke problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Kamke_problem&oldid=16908
Hilbert-Kamke problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Kamke_problem&oldid=16908
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article