On the exponents of the convergence of singular integrals and singular series of a multivariate problem
Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 46-57
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In the paper we continue studies on the theory of multivariate trigonometric sums, in the base of which lies of the I.M.Vinogradov's method. Here we obtain for $n=r=2$ lower estimates of the convergence exponent of the singular series and the singular integral of the asymptotic formulas for $P\to\infty$ for the number of solutions of the following system of Diophantine equations $$ \sum_{j=1}^{2k}(-1)^jx_{1,j}^{t_1}\dots x_{r,j}^{t_r}=0, 0\leq t_1,\dots, t_r\leq n, $$ where $n\geq 2,r\geq 1, k$ are natural numbers, moreover an each variable $x_{i,j}$ can take all integer values from $1$ to $P\geq 1.$
Keywords:
exponent of the convergence, singular integrals, singular series.
@article{CHEB_2019_20_4_a3,
author = {L. G. Arkhipova and V. N. Chubarikov},
title = {On the exponents of the convergence of singular integrals and singular series of a multivariate problem},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {46--57},
publisher = {mathdoc},
volume = {20},
number = {4},
year = {2019},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/}
}
TY - JOUR AU - L. G. Arkhipova AU - V. N. Chubarikov TI - On the exponents of the convergence of singular integrals and singular series of a multivariate problem JO - Čebyševskij sbornik PY - 2019 SP - 46 EP - 57 VL - 20 IS - 4 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/ LA - ru ID - CHEB_2019_20_4_a3 ER -
%0 Journal Article %A L. G. Arkhipova %A V. N. Chubarikov %T On the exponents of the convergence of singular integrals and singular series of a multivariate problem %J Čebyševskij sbornik %D 2019 %P 46-57 %V 20 %N 4 %I mathdoc %U http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/ %G ru %F CHEB_2019_20_4_a3
L. G. Arkhipova; V. N. Chubarikov. On the exponents of the convergence of singular integrals and singular series of a multivariate problem. Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 46-57. http://geodesic.mathdoc.fr/item/CHEB_2019_20_4_a3/