In this paper, we use the elementary and analytic methods to investigate the computational problem of quadratic character sums related to quaternary and quintuple symmetric polynomials modulo a prime $ p $, respectively. By applying properties of Gauss sums and those of third-order characters modulo $ p $, we establish some identities for such quadratic character sums in the cases of $ (3, p-1) = 1 $ and $ (3, p-1) = 3 $.
Citation: Shaofan Cao, Tingting Wang. On quadratic character sums of multivariate symmetric polynomials modulo $ p $[J]. Electronic Research Archive, 2025, 33(12): 7491-7508. doi: 10.3934/era.2025330
In this paper, we use the elementary and analytic methods to investigate the computational problem of quadratic character sums related to quaternary and quintuple symmetric polynomials modulo a prime $ p $, respectively. By applying properties of Gauss sums and those of third-order characters modulo $ p $, we establish some identities for such quadratic character sums in the cases of $ (3, p-1) = 1 $ and $ (3, p-1) = 3 $.
| [1] | T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York-Heidelberg, 1976. |
| [2] | L. K. Hua, Introduction to Number Theory, Science Publishing Co., Peking, 1957. |
| [3] | K. F. Ireland, M. I. Rosen, A Classical Introduction to Modern Number Theory, Grad. Texts in Math., 84, Springer-Verlag, New York-Berlin, 1982. |
| [4] |
X. G. Liu, Y. Y. Meng, On the quadratic residues and their distribution properties, Open Math., 21 (2023), 20230123. https://doi.org/10.1515/math-2023-0123 doi: 10.1515/math-2023-0123
|
| [5] |
C. K. Ren, Z. W. Sun, On some determinants arising from quadratic residues, Bull. Iranian Math. Soc, 51 (2025), 59. https://doi.org/10.1007/s41980-025-00982-4 doi: 10.1007/s41980-025-00982-4
|
| [6] |
X. Wang, H. Fang, The distribution properties of consecutive quadratic residue sequences, J. Math., (2023), 5253261. https://doi.org/10.1155/2023/5253261 doi: 10.1155/2023/5253261
|
| [7] |
T. T. Wang, X. X. Lv, The quadratic residues and some of their new distribution properties, Symmetry, 12 (2020), 421. https://doi.org/10.3390/sym12030421 doi: 10.3390/sym12030421
|
| [8] |
A. Weil, On some exponential sums, Proc. Nat. Acad. Sci. U.S.A., 34 (1948), 204–207. https://doi.org/10.1073/pnas.34.5.204 doi: 10.1073/pnas.34.5.204
|
| [9] |
Y. He, Q. Y. Liao, On an identity associated with Weil's estimate and its applications, J. Number Theory, 129 (2009), 1075–1089. https://doi.org/10.1016/j.jnt.2008.10.022 doi: 10.1016/j.jnt.2008.10.022
|
| [10] |
B. Nilanjan, R. L. Antonio, W. P. Zhang, An explicit evaluation of 10th-power moment of quadratic Gauss sums and some applications, Funct. Approx. Comment. Math., 66 (2022), 253–274. https://doi.org/10.7169/facm/1995 doi: 10.7169/facm/1995
|
| [11] |
Y. Y. Meng, On a certain quadratic character sums of ternary symmetry polynomials mod $p$, J. Math., (2021), 5572835. https://doi.org/10.1155/2021/5572835 doi: 10.1155/2021/5572835
|
| [12] |
S. Chowla, J. Cowles, M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory, 9 (1977), 502–506. https://doi.org/10.1016/0022-314X(77)90010-5 doi: 10.1016/0022-314X(77)90010-5
|
| [13] |
B. C. Berndt, R. J. Evans, Sums of Gauss, Jacobi, and Jacobsthal, J. Number Theory, 11 (1979), 349–398. https://doi.org/10.1016/0022-314X(79)90008-8 doi: 10.1016/0022-314X(79)90008-8
|
| [14] | W. P. Zhang, J. Y. Hu, The number of solutions of the diagonal cubic congruence equation mod $p$, Math. Rep. (Bucur.), 20 (2018), 73–80. |
| [15] |
B. C. Berndt, R. J. Evans, The determination of Gauss sum, Bull. Amer. Math. Soc. (N.S.), 5 (1981), 107–129. https://doi.org/10.1090/S0273-0979-1981-14930-2 doi: 10.1090/S0273-0979-1981-14930-2
|
| [16] |
G. H. Chen, W. P. Zhang, One kind two-term exponential sums weighted by third-order character, AIMS Math., 9 (2024), 9597–9607. https://doi.org/10.3934/math.2024469 doi: 10.3934/math.2024469
|
Shaofan Cao, Tingting Wang. On quadratic character sums of multivariate symmetric polynomials modulo $ p $[J]. Electronic Research Archive, 2025, 33(12): 7491-7508. doi: 10.3934/era.2025330
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