Literature cited
E. Waring, Meditationes Algebraicae, Cambridge (1770).
D. Hilbert, “Beweis für die Darstellbarkeit der ganzen Zahlen durch eine feste Anzahl n-ter Potenzen,” Math. Ann.,67, 281–300 (1909).
G. H. Hardy and J. E. Littlewood, “Some problems of ‘partitio numerorum,’ I: A new solution of Waring's problem,” Nachr. Akad. Wiss. Gottingen Math.-Phys. K1., 33–54 (1920); “IV: The singular series in Waring's problem and the value of the number G(k),” Math. Z., 12, 161–168 (1922).
I. M. Vinogradov, “Sur un théorème général de Waring,” Mat. Sb.,31, 490–507 (1924).
I. M. Vinogradov, “On an upper bound for G(n) in Waring's problem,” Izv. Akad. Nauk SSSR,5, No. 6, 1455–1469 (1934).
I. M. Vinogradov, “On the question of an upper bound for G(n),” Izv. Akad. Nauk SSSR Ser. Mat.,23, No. 5, 637–642 (1959).
A. A. Karatsuba, “Waring's problem for congruences modulo a prime power,” Vestn. Mosk. Gos. Univ.,1, No. 4, 28–38 (1962).
A. A. Karatsuba, “Trigonometric sums of a special form and their applications,” Izv. Akad. Nauk SSSR Ser. Mat.,28, No. 1, 237–248 (1964).
A. A. Karatsuba, “On systems of congruences,” Izv. Akad. Nauk SSSR Ser. Mat.,29, No. 4, 935–944 (1965).
A. A. Karatsuba, “On an estimate of the number of solutions of certain equations,”Dokl. Akad. Nauk SSSR,165, No. 1, 31–32 (1965).
A. A. Karatsuba, “Systems of congruences and equations of Waring type,” Dokl. Akad. Nauk SSSR,165., No. 2, 274–276 (1965).
A. A. Karatsuba, “Theorems on the mean and complete trigonometric sums,” Izv. Akad. Nauk SSSR Ser. Mat.,30, No. 2, 183–206 (1966).
A. A. Karatsuba, “On some arithmetical problems involving numbers having small prime divisors,” Acta Arith.,27, 489–492 (1975).
A. A. Karatsuba, “Trigonometric sums and their applications,” Proceedings of the International Congress of Mathematicians, Vol. I, Vancouver (1975), pp. 365–368.
A. A. Karatsuba, “On a system of congruences,” Mat. Zametki,19, No. 3, 389–392 (1976).
A. A. Karatsuba, “On the function G(n) in Waring's problem,” Izv. Akad. Nauk SSSR Ser. Mat.,49, No. 5, 935–947 (1985).
I. M. Vinogradov, The Method of Trigonometric Sums in Number Theory [in Russian], 2nd edn., Nauka, Moscow (1980).
B. M. Bredikhin and T. I. Grishina, “An elementary estimate of G(n) in Waring's problem,” Mat. Zametki,24, No. 1, 7–18 (1978).
L.-K. Hua, The Method of Trigonometric Sums and Its Applications in Number Theory [Russian translation], Mir, Moscow (1964).
R. C. Vaughan, “A survey of some important problems in additive number theory,” Société Math. France, Astérisque,61, 213–222 (1979).
E. Kamke, “Verallgemeinerungen des Waring-Hilbertschen Satzes,” Math. Ann.,3, 3–38 (1921).
K. K. Mardzhanishvili, “On the simultaneous representation of n numbers by sums of perfect first, second, ..., n-th powers,“ Izv. Akad. Nauk SSSR Ser. Mat.,1, 609–631 (1937).
K. K. Mardzhanishvili, “On a system of Diophantine equations,” Dokl. Akad. Nauk SSSR,22, No. 11, 471–474 (1939).
K. K. Mardzhanishvili, “On a problem of additive number theory,” Izv. Akad. Nauk SSSR Ser. Mat., No. 4, 193–214 (1940).
K. K. Mardzhanishvili, “On certain nonlinear systems of equations in integers,” Mat. Sb.,32(75), No. 3, 630–675 (1953).
K. K. Mardzhanishvili, “On a singular series,” Trudy Mat. Inst. Akad. Nauk,142, 174–181 (1976).
G. I. Arkhipov, “On the value of the singular series in the Hilbert-Kamke problem,” Dokl. Akad. Nauk SSSR,259, No. 2, 265–267 (1981).
I. M. Vinogradov, “Some problems of analytic number theory,” Proceedings of the Third All-Union Mathematical Congress, Moscow (1958), pp. 3–13.
Yu. V. Linnik, “Some remarks on estimates of trigonometric sums,” Usp. Mat. Nauk,14, No. 3, 153–160 (1959).
G. I. Arkhipov, “On the Hilbert-Kamke problem,” Izv. Akad. Nauk SSSR Ser. Mat.,48, No. 1, 3–52 (1984).
G. I. Arkhipov, “Investigations on the Hilbert-Kamke problem,” Doctoral Dissertation, Mathematical Institute of the Academy of Sciences, Moscow (1984).
V. N. Chubarikov, “On the simultaneous representation of natural numbers by sums of prime powers,” Dokl. Akad. Nauk SSSR,286, No. 4, 864–867 (1986).
V. B. Dem'yanov, “On cubic forms in discretely valuated fields,” Dokl. Akad. Nauk SSSR,74, No. 5, 889–891 (1950).
D. J. Lewis, “Cubic homogeneous polynomials over p-adic number fields,” Ann. Math.,56, No. 3, 473–478 (1952).
Yu. L. Ershov, “On elementary theories of local fields,” Algebra Logika,4, No. 2, 5–30 (1965).
J. Ax and S. Kochen, “Diophantine problems over local fields,” Am. J. Math.,57, No. 3, 605–630 (1965).
G. Terjanian, “Un contre-exemple à une conjecture d'Artin,” Compt. Rend. Acad. Sci.,262, 612 (1966).
J. Browkin, “On forms over p-adic fields,” Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.,14, No. 9, 489–492 (1966).
G. I. Arkhipov and A. A. Karatsuba, “On local representation of zero by a form,” Izv. Akad. Nauk SSSR Ser. Mat.,45, No. 3, 948–961 (1981).
G. I. Arkhipov and A. A. Karatsuba, “On the representation of zero by a form in the field of p-adic numbers,“ Dokl. Akad. Nauk SSSR,262, No. 1, 11–13 (1982).
G. I. Arkhipov and A. A. Karatsuba, “On a problem of the theory of congruences,” Usp. Mat. Nauk,37, No. 5 (227), 161–162 (1982).
G. I. Arkhipov and V. N. Chubarikov, “On arithmetical conditions for the solvability of nonlinear systems of Diophantine equations,” Dokl. Akad. Nauk SSSR,284, No. 1, 16–21 (1985).
V. N. Chubarikov, “Multiple trigonometric sums with prime numbers,” Dokl. Akad. Nauk SSSR,278, No. 2, 302–304 (1984).
V. N. Chubarikov, “Estimates of multiple trigonometric sums with prime numbers,” Izv. Akad. Nauk SSSR Ser. Mat.,49, No. 5, 1031–1067 (1985).
G. I. Arkhipov, “Multiple trigonometric sums,” Dokl. Akad. Nauk SSSR,219, No. 5, 1036–1037 (1974).
G. I. Arkhipov, “A theorem on the mean value of the modulus of a multiple trigonometric sum,” Mat. Zametki,17, No. 1, 143–153 (1975).
G. I. Arkhipov, “Estimates of H. Weyl's double trigonometric sums,” Trudy Mat. Inst. Akad. Nauk,142, 46–66 (1976).
G. I. Arkhipov and V. N. Chubarikov, “On multiple trigonometric sums,” Dokl. Akad. Nauk SSSR,222, No. 5, 1017–1019 (1975).
G. I. Arkhipov and V. N. Chubarikov,“Multiple trigonometric sums,” Izv. Akad. Nauk SSSR Ser. Mat.,40, No. 1, 209–220 (1976).
V. N. Chubarikov, “On a multiple trigonometric integral,” Dokl. Akad. Nauk SSSR,227, No. 6, 1308–1310 (1976).
V. N. Chubarikov, “On multiple rational trigonometric sums and multiple integrals,” Mat. Zametki,20, No. 1, 61–68 (1976).
V. N. Chubarikov, “An asymptotic formula forthe mean value of a multiple trigonometric sum,” Mat. Zametki,23, No. 6, 799–816 (1978).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “An upper bound for the modulus of a multiple trigonometric sum,” Trudy Mat. Inst. Akad. Nauk,143, 3–31 (1977).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov,“A sharp estimate of the number of solutions of a system of Diophantine equations,” Izv. Akad. Nauk SSSR Ser. Mat., No. 6, 1187–1226 (1978).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Distribution of fractional parts of polynomials in several variables,” Mat. Zametki,25, No. 1, 3–14 (1979).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Multiple trigonometric sums,” Trudy Mat. Inst. Akad. Nauk,151, 1–128 (1980).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “On a system of Diophantine equations,” Dokl. Akad. Nauk SSSR,252, No. 2, 275–276 (1980).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Uniform estimates of multiple trigonometric sums,” Dokl. Akad. Nauk SSSR,252, No. 6, 1289–1391 (1980).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Multiple trigonometric sums and their applications,” Izv. Akad. Nauk SSSR Ser. Mat.,44, No. 4, 723–781 (1980).
V. N. Chubarikov, “On asymptotic formulas for I. M. Vinogradov's integral and its generalizations,” Trudy Mat. Inst. Akad. Nauk,157, 214–232 (1981).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “New uniform estimates of multiple trigonometric sums,” Dokl. Akad. Nauk SSSR,272, No. 1, 11–12 (1983).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Particular cases of the theory of multiple trigonometric sums,” Izv. Akad. Nauk SSSR Ser. Mat.,47, No. 4, 707–784 (1983).
V. N. Chubarikov, “On a multiple trigonometric integral,” Dokl. Akad. Nauk SSSR,227, No. 6, 1308–1310 (1976).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Trigonometric integrals,” Izv. Akad. Nauk SSSR Ser. Mat.,43, No. 5, 971–1003 (1979).
G. I. Arkhipov, A. A. Karatsuba, and V. N. Chubarikov, “Exponent of convergence of the singular integral of Tarry's problem,” Dokl. Akad. Nauk SSSR,248, No. 2, 268–272 (1979).
L.-K. Hua, “On Tarry's problem,” Proceedings of the Third All-Union Mathematical Congress, Vol. IV, Moscow (1959), pp. 140–143.
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Translated from Matematicheskie Zametki, Vol. 41, No. 2, pp. 272–284, February, 1987.
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Karatsuba, A.A. The Hilbert-Kamke problem in analytic number theory. Mathematical Notes of the Academy of Sciences of the USSR 41, 155–161 (1987). https://doi.org/10.1007/BF01138339
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DOI: https://doi.org/10.1007/BF01138339