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Difference between revisions of "Hilbert-Kamke problem"

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The problem of the compatibility of a system of [[Diophantine equations|Diophantine equations]] of Waring type:
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The problem of the compatibility of a system of [[Diophantine equations]] of [[Waring problem|Waring type]]:
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\begin{equation}\label{eq:1}
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\left.{
 +
\begin{array}{rcl}
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x_1^n + \cdots + x_s^n &=& N_n \\
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x_1^{n-1} + \cdots + x_s^{n-1} &=& N_{n-1} \\
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\ldots&&\\
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x_1 + \cdots + x_s &=& N_1
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\end{array}
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}\right\rbrace
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\end{equation}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047290/h0472901.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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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.
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047290/h0472902.png" /> assume integral non-negative values, certain additional restrictions [[#References|[3]]] are imposed on the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047290/h0472903.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047290/h0472904.png" /> is a sufficiently-large number which depends only on the natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047290/h0472905.png" /> which is given in advance.
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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.
  
The Hilbert–Kamke problem, which was posed in 1900 by D. Hilbert [[#References|[1]]], was solved by E. Kamke, who proved that solutions to (*) 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|Vinogradov method]] for estimating trigonometric sums.
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====References====
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<table>
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<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</TD></TR>
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<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)</TD></TR>
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<TR><TD valign="top">[3]</TD> <TD valign="top">  K.K. Mardzhanishvili,  ''Izv. Akad. Nauk SSSR Ser. Mat.''  (1937)  pp. 609–631</TD></TR>
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</table>
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,  "Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047290/h0472906.png" /> Potenzen (Waringsches Problem)"  ''Math. Ann.'' , '''67'''  (1909)  pp. 281–300</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)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.K. Mardzhanishvili,  ''Izv. Akad. Nauk SSSR Ser. Mat.''  (1937)  pp. 609–631</TD></TR></table>
 

Revision as of 19:04, 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
[2] I.M. Vinogradov, "The method of trigonometric sums in the theory of numbers" , Interscience (1954) (Translated from Russian)
[3] K.K. Mardzhanishvili, Izv. Akad. Nauk SSSR Ser. Mat. (1937) pp. 609–631
How to Cite This Entry:
Hilbert-Kamke problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Kamke_problem&oldid=22569
This article was adapted from an original article by B.M. Bredikhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article