On a system of comparisons for polynomials in several variables
https://doi.org/10.22405/2226-8383-2026-27-2-212-218
Abstract
In this paper the solvability of a multidimensional additive system of congruences prime modulo for polynomials of several variables is proved. A.A.Karatsuba completely studied the one-dimensional case, getting the right order of the number of summands for solvability of the considering system of congruences.
About the Author
Vladimir Nikolaevich ChubarikovRussian Federation
doctor of physical and mathematical sciences, professor
References
1. Vinogradov, I.M. 1933, “On problems of analytic number theory” [O problemakh analiticheskoi teorii chisel], in Proceedings of the November Jubilee Session of the USSR Academy of Sciences, Moscow.
2. Vinogradov, I.M. 1980, Method of trigonometric sums in number theory [Metod trigonometricheskikh summ v teorii chisel], 2nd edn, Nauka, Moscow.
3. Vinogradov, I.M. 1936, “A new method of estimation of trigonometrical sums”, Mathematics of the USSR-Sbornik, vol. 43, pp. 175–188.
4. Karatsuba, A.A. 1976, “On a system of congruences” [Ob odnoi sisteme sravnenii], Matematicheskie Zametki, vol. 19, no. 3, pp. 389–391.
5. Mardzhanishvili, K.K. 1937, “On the simultaneous representation of numbers by sums of full first, second, . . . , 𝑛-th powers” [Ob odnovremennom predstavlenii chisel summami polnykh pervykh, vtorykh, . . . , 𝑛-kh stepenei], Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 1, no. 2, pp. 609–631.
6. Mardzhanishvili, K.K. 1950, “On a system of equations in prime numbers” [Ob odnoi sisteme uravnenii v prostykh chislakh], Doklady Akademii Nauk SSSR, vol. 70, pp. 381–383.
7. Mardzhanishvili, K.K. 1953, “On some nonlinear systems of equations in integers” [O nekotorykh nelineinykh sistemakh uravnenii v tselykh chislakh], Matematicheskii Sbornik, vol. 33, pp. 630–675.
8. Linnik, Yu.V. 1953, “Some remarks on estimates of trigonometric sums” [Nekotorye zamechaniya ob otsenkakh trigonometricheskikh summ], Uspekhi Matematicheskikh Nauk, vol. 33, pp. 630–675.
9. Arkhipov, G.I. 1975, “Mean value theorem for the modulus of a multiple trigonometric sum” [Teorema o srednem znachenii modulya kratnoi trigonometricheskoi summy], Matematicheskie Zametki, vol. 17, pp. 84–90.
10. Arkhipov, G.I. & Chubarikov, V.N. 1976, “Multiple trigonometric sums” [Kratnye trigonometricheskie summy], Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 40, no. 1, pp. 209–220.
11. Hua, L.-K. 1940, “On an exponential sums”, Journal of the Chinese Mathematical Society, vol. 2, pp. 301–312.
12. Chen, J.-R. 1977, “On Professor Hua’s estimate on exponential sums”, Acta Scientiarum Sinica, vol. 20, no. 6, pp. 711–719.
13. Chubarikov, V.N. 1976, “On multiple rational trigonometric sums and multiple integrals” [O kratnykh ratsionalnykh trigonometricheskikh summakh i kratnykh integralakh], Matematicheskie Zametki, vol. 20, pp. 589–593.
14. Arkhipov, G.I., Karatsuba, A.A. & Chubarikov, V.N. 1979, “Trigonometric integrals” [Trigonometricheskie integraly], Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 43, no. 5, pp. 971–1003.
15. Arkhipov, G.I., Chubarikov, V.N. & Karatsuba, A.A. 2004, Trigonometric sums in number theory and analysis, De Gruyter Expositions in Mathematics, vol. 39, Walter de Gruyter, Berlin – New York.
16. Hua, L.-K. 1947, “Additive theory of prime numbers”, Trudy Matematicheskogo Instituta imeni V.A. Steklova, vol. 22, pp. 1–179.
17. Hua, L.-K. 1953, “On the number of solutions of Tarry’s problem”, Acta Scientiarum Sinica, vol. 1, pp. 1–76.
18. Chubarikov, V.N. 1981, “On asymptotic formulas for the I.M. Vinogradov integral and its generalizations” [Ob asimptoticheskikh formulakh dlya integrala I.M. Vinogradova i ego obobshchenii], Trudy Matematicheskogo Instituta imeni V.A. Steklova, vol. 157, pp. 214–232.
19. Chubarikov, V.N. 1985, “Estimates of multiple trigonometric sums with prime numbers” [Otsenki kratnykh trigonometricheskikh summ s prostymi chislami], Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 49, no. 5, pp. 1031–1067.
Review
For citations:
Chubarikov V.N. On a system of comparisons for polynomials in several variables. Chebyshevskii Sbornik. 2026;27(2):212-218. (In Russ.) https://doi.org/10.22405/2226-8383-2026-27-2-212-218
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