Geodesic
Exact value of the exponent of~convergence of~the~singular~integral in Tarry's problem for~homogeneous polynomials of degree~$n$ in two variables
Izvestiya. Mathematics , Tome 85 (2021) no. 2, pp. 332-340.

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Jabbarov [1] obtained the exact value of the exponent of convergence of the singular integral in Tarry's problem for homogeneous polynomials of degree $2$. We extend this result to the case of polynomials of degree $n$.
Keywords: oscillatory integrals, singular integral, Tarry's problem. 
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M. A. Chahkiev. Exact value of the exponent of~convergence of~the~singular~integral in Tarry's problem for~homogeneous polynomials of degree~$n$ in two variables. Izvestiya. Mathematics , Tome 85 (2021) no. 2, pp. 332-340. http://geodesic.mathdoc.fr/item/IM2_2021_85_2_a6/

[1] I. Sh. Jabbarov, “Convergence exponent of a special integral in the two-dimensional Tarry problem with homogeneous polynomial of degree 2”, Math. Notes, 105:3 (2019), 359–365 | DOI | DOI | MR | Zbl

[2] Loo-Keng Hua, “On the number of solutions of Tarry's problem”, Acta Sci. Sinica, 1:1 (1952), 1–76 | MR

[3] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, “Multiple trigonometric sums”, Proc. Steklov Inst. Math., 151:2 (1982), 1–126 | MR | Zbl

[4] G. I. Arkhipov, A. A. Karatsuba, V. N. Chubarikov, “The index of convergence of the singular integral in Tarry's problem”, Soviet Math. Dokl., 20:5 (1979), 978–981 | MR | Zbl

[5] G. I. Arkhipov, V. N. Chubarikov, A. A. Karatsuba, Trigonometric sums in number theory and analysis, De Gruyter Exp. Math., 39, Walter de Gruyter GmbH Co. KG, Berlin, 2004, x+554 pp. | DOI | MR | MR | Zbl | Zbl

[6] G. I. Arhipov, A. A. Karatsuba, V. N. Chubarikov, “Trigonometric integrals”, Math. USSR-Izv., 15:2 (1980), 211–239 | DOI | MR | Zbl

[7] I. A. Ikromov, “On the convergence exponent of trigonometric integrals”, Analiticheskaya teoriya chisel i prilozheniya, Sbornik statei. K 60-letiyu so dnya rozhdeniya professora Anatoliya Alekseevicha Karatsuby, Tr. MIAN, 218, Nauka, M., 1997, 179–189 | MR | Zbl

[8] M. A. Chakhkiev, “On the convergence exponent of the singular integral in the multi-dimensional analogue of Tarry's problem”, Izv. Math., 67:2 (2003), 405–418 | DOI | DOI | MR | Zbl

[9] M. A. Chakhkiev, “Estimates for oscillatory integrals with convex phase”, Izv. Math., 70:1 (2006), 171–209 | DOI | DOI | MR | Zbl

[10] A. Zygmund, Trigonometric series, v. II, 2nd ed., Cambridge Univ. Press, New York, 1959, vii+354 pp. | MR | MR | Zbl | Zbl