Geodesic
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

Voir la notice de l'article provenant de la source Math-Net.Ru

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/