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Zapiski Nauchnykh Seminarov LOMI, 1981, Volume 109, Pages 3–33
(Mi znsl3917)
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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.
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
Linking options:
https://www.mathnet.ru/eng/znsl3917 https://www.mathnet.ru/eng/znsl/v109/p3
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Abstract page: | 231 | Full-text PDF : | 58 |
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