On Hua Loo-Keng’s Estimates of Exponential Sums in Algebraic Number Fields
https://doi.org/10.22405/2226-8383-2024-25-2-181-207
Abstract
This paper provides a generalization of the Hua Loo-Keng estimation method of rational trigonometric sums with a polynomial in exponent in algebraic number fields, which are extensions of the field of rational numbers. In the ring of integers of this algebraic number field we consider integer and fractional ideals. For a complete system of residues for any integer ideal, Hua Loo-Keng proved an analogue of the Euler–Fourier formula, which, using results
regarding the multiplicity of roots of a polynomial congruence modulo a prime ideal (“Hua Loo-Keng trees”), allows the problem to be reduced to the p-adic lifting of solutions, and this allows us to reduce the problem of estimating the sum to estimating the number of solutions of polynomial congruences modulo a power of a prime ideal. Furthermore, building upon Chen Jingrun’s estimates in the field of rational numbers, we obtain improved constants for similar estimates in algebraic numeric fields.
References
1. Chubarikov, V. N., 2021. “On multiple rational trigonometric sums over a field of algebraic
2. numbers”, Chebyshevskii Sbornik, Vol. 22, Iss. 4, pp. 306–323.
3. Hua, L-K., 1951. “On Exponential Sums Over an Algebraic Number Field”, Canadian Journal of Mathematics, 3:44-51. doi:10.4153/CJM-1951-006-4.
4. Hua, L-K., 1952. “On the number of solutions of Tarry’s problem”, Acta Sci. Sinica, Vol. 1. pp. 1-–76.
5. Hua, L-K., 1938. “On An Exponential Sum”, Journal of the London Mathematical Society, s1-13:54-61. https://doi.org/10.1112/jlms/s1-13.1.54.
6. Arkhipov, G. I., Karatsuba, A. A., Chubarikov, V. N., 1987. “Theory of multiple trigonometric
7. sums”, M.: Nauka. Fizmatlit.
8. Wang Yuan, 1991. “Diophantine Equations and Inequalities in Algebraic Number Fields”,
9. Springer Verlag, Berlin, Heidelberg.
10. Anthony Knapp, 2006. “Advanced Algebra”, Birkh¨auser Boston.
11. Anthony Knapp, 2006. “Basic Algebra”, Birkh¨auser Boston, 2006.
12. Weil, A., 1948. “On Some Exponential Sums”, Proceedings of the National Academy of Sciences of the United States of America, 34(5), pp. 204–207. http://www.jstor.org/stable/88420.
13. Chubarikov, V. N., 2007. “Hua Lo-ken trees in the theory of comparisons”, Mathematical issues of cybernetics, Iss. 16. — M.: FIZMATLIT, pp. 73–78.
14. Chen Jingrun, 1977. “On Professor Hua’s Estimate of Exponential Sums”, Scientia Sinica, 20, 6: pp. 711-719.
15. Chen Jingrun, 1959. “On the representation of natural number as a sum of terms of the form 𝑥(𝑥+1)···(𝑥+𝑘−1)/𝑘! ”, Acta Mathematica Sinica, pp. 264–270.
16. Nechaev, V. I., 1975. “An estimate of the complete rational trigonometric sum”, Math. Notes, 17, № . 6, pp. 504–511.
17. Qi Minggao & Ding Ping, 1985. “On Estimate of complete trigonometric sums”, China Ann.
18. Math. Vol. 6, pp. 109–120.
19. Mordell, L. J., 1932. “On a sum analogous to a Gauss’s sum”, Quart. J. Math. (Oxford), Vol. 3, pp. 161-–167.
Review
For citations:
Hafez A. On Hua Loo-Keng’s Estimates of Exponential Sums in Algebraic Number Fields. Chebyshevskii Sbornik. 2024;25(2):181-207. (In Russ.) https://doi.org/10.22405/2226-8383-2024-25-2-181-207
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