A note on some diagonal cubic equations over finite fields

Wenxu Ge; Weiping Li; Tianze Wang; Wenxu Ge; Weiping Li; Tianze Wang

doi:10.3934/math.20241053

AIMS Mathematics

2024, Volume 9, Issue 8: 21656-21671. doi: 10.3934/math.20241053

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A note on some diagonal cubic equations over finite fields

1.
School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450046, China
2.
School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou 450046, China
3.
Institute of Mathematics, Henan Academy of Sciences, Zhengzhou 450046, China

Received: 10 May 2024 Revised: 22 June 2024 Accepted: 28 June 2024 Published: 08 July 2024
MSC : 11T23, 11T24

Let a prime $ p\equiv 1(\text{mod}3) $ and $ z $ be non-cubic in $ \mathbb{F}_p $. Gauss proved that the number of solutions of equation
$ x_1^3+x_2^3+zx_3^3 = 0 $
in $ \mathbb{F}_p $ was $ p^2+\frac{1}{2}(p-1)(9d-c) $, where $ c $ was uniquely determined and $ d $, except for the sign, was defined by
$ 4p = c^2+27d^2,\ \ c\equiv 1(\text{mod}3). $
In 1978, Chowla, Cowles, and Cowles determined the sign of $ d $ for the case of 2 being non-cubic in $ \mathbb{F}_p $. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field $ \mathbb{F}_q $ with $ q = p^k $, $ p\equiv 1(\text{mod}3) $, and determined the sign of $ d $ for the case of 3 being non-cubic.
- Gauss sum,
- finite field,
- diagonal cubic equation
Citation: Wenxu Ge, Weiping Li, Tianze Wang. A note on some diagonal cubic equations over finite fields[J]. AIMS Mathematics, 2024, 9(8): 21656-21671. doi: 10.3934/math.20241053

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Abstract

Let a prime $ p\equiv 1(\text{mod}3) $ and $ z $ be non-cubic in $ \mathbb{F}_p $. Gauss proved that the number of solutions of equation

$ x_1^3+x_2^3+zx_3^3 = 0 $

in $ \mathbb{F}_p $ was $ p^2+\frac{1}{2}(p-1)(9d-c) $, where $ c $ was uniquely determined and $ d $, except for the sign, was defined by

$ 4p = c^2+27d^2,\ \ c\equiv 1(\text{mod}3). $

In 1978, Chowla, Cowles, and Cowles determined the sign of $ d $ for the case of 2 being non-cubic in $ \mathbb{F}_p $. In this paper, we extended the result of Chowla, Cowles and Cowles to finite field $ \mathbb{F}_q $ with $ q = p^k $, $ p\equiv 1(\text{mod}3) $, and determined the sign of $ d $ for the case of 3 being non-cubic.

References

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