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Zapiski Nauchnykh Seminarov LOMI, 1981, Volume 109, Pages 3–33 (Mi znsl3917)  

This article is cited in 3 scientific papers (total in 3 papers)

System of inequalities

V. A. Bykovskii
Abstract: Let $N_{k,n,r}(P)$ be a number of integer solutions of the system of inequalities
$$ |x_1^\nu+\dots+x_k^\nu-y_1^\nu-\dots-y_k^\nu|\le P^{\nu-r},\ \ 1\le\nu\le n;\quad1\le x_1,\dots,x_k,y_1,\dots,y_k\le P. $$
The main result is the following estimate for $k-\frac{n^2}4\gg nr\log r$
$$ N_{k,n,r}(P)\ll P^{2k-\frac{n(n+1)}2+\frac{(n-r)(n-r+1)}2}. $$
This estimate has the right order with respect to $P$. For $r=n$ this is the classical Vinogradov mean value theorem.
English version:
Journal of Soviet Mathematics, 1984, Volume 24, Issue 2, Pages 159–178
DOI: https://doi.org/10.1007/BF01087239
Bibliographic databases:
Document Type: Article
UDC: 511.292
Language: Russian
Citation: V. A. Bykovskii, “System of inequalities”, Differential geometry, Lie groups and mechanics. Part IV, Zap. Nauchn. Sem. LOMI, 109, "Nauka", Leningrad. Otdel., Leningrad, 1981, 3–33; J. Soviet Math., 24:2 (1984), 159–178
Citation in format AMSBIB
\Bibitem{Byk81}
\by V.~A.~Bykovskii
\paper System of inequalities
\inbook Differential geometry, Lie groups and mechanics. Part~IV
\serial Zap. Nauchn. Sem. LOMI
\yr 1981
\vol 109
\pages 3--33
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl3917}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=629113}
\zmath{https://zbmath.org/?q=an:0534.10030|0472.10039}
\transl
\jour J. Soviet Math.
\yr 1984
\vol 24
\issue 2
\pages 159--178
\crossref{https://doi.org/10.1007/BF01087239}
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  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Записки научных семинаров ПОМИ
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