Home Mathematics On the number of zeros of diagonal quartic forms over finite fields
Article
Licensed
Unlicensed Requires Authentication

On the number of zeros of diagonal quartic forms over finite fields

  • Junyong Zhao , Yulu Feng , Shaofang Hong EMAIL logo and Chaoxi Zhu
Published/Copyright: January 28, 2022
Forum Mathematicum
From the journal Forum Mathematicum Volume 34 Issue 2

Abstract

Let 𝔽q be the finite field of q=pm≡1(mod4) elements with p being an odd prime and m being a positive integer. For c,y∈𝔽q with y∈𝔽q* non-quartic, let Nn⁢(c) and Mn⁢(y) be the numbers of zeros of x14+⋯+xn4=c and x14+⋯+xn-14+y⁢xn4=0, respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function ∑n=1∞Nn⁢(0)⁢xn is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions

∑n=1∞Nn⁢(c)⁢xn and ∑n=1∞Mn+1⁢(y)⁢xn

are rational functions in x. We also obtain the explicit expressions of these generating functions. Our result extends Myerson’s theorem gotten in 1979.

Keywords: Diagonal quartic form; cyclotomic number; generating function; finite fields
MSC 2010: 11T06; 11T22

Communicated by Freydoon Shahidi

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12171332

Funding statement: S. F. Hong was supported partially by National Science Foundation of China Grant no. 12171332.

Acknowledgements

The authors would like to thank Professor Freydoon Shahidi and the anonymous referee for their careful reading of the manuscript and helpful suggestions that improve the presentation of the paper.

References

[1] A. Adolphson and S. Sperber, p-adic estimates for exponential sums and the theorem of Chevalley–Warning, Ann. Sci. Éc. Norm. Supér. (4) 20 (1987), no. 4, 545–556. 10.24033/asens.1543Search in Google Scholar

[2] T. M. Apostol, Introduction to Analytic Number Theory, Undergrad. Texts Math., Springer, New York, 1976. 10.1007/978-1-4757-5579-4Search in Google Scholar

[3] J. Ax, Zeroes of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261. 10.2307/2373163Search in Google Scholar

[4] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, Canad. Math. Soc. Ser. Monogr. Adv. Texts, John Wiley & Sons, New York, 1998. Search in Google Scholar

[5] W. Cao, A partial improvement of the Ax–Katz theorem, J. Number Theory 132 (2012), no. 4, 485–494. 10.1016/j.jnt.2011.12.002Search in Google Scholar

[6] C. Chevalley, Démonstration d’une hypothèse de M. Artin, Abh. Math. Semin. Univ. Hambg. 11 (1935), no. 1, 73–75. 10.1007/BF02940714Search in Google Scholar

[7] S. Chowla, J. Cowles and M. Cowles, On the number of zeros of diagonal cubic forms, J. Number Theory 9 (1977), no. 4, 502–506. 10.1016/0022-314X(77)90010-5Search in Google Scholar

[8] S. Chowla, J. Cowles and M. Cowles, The number of zeroes of x3+y3+c⁢z3 in certain finite fields, J. Reine Angew. Math. 299(300) (1978), 406–410. 10.1515/crll.1978.299-300.406Search in Google Scholar

[9] C. F. Gauss, Disquisitiones Arithmeticae, Yale University, New Haven, 1966. Search in Google Scholar

[10] S. F. Hong and C. X. Zhu, On the number of zeros of diagonal cubic forms over finite fields, Forum Math. 33 (2021), no. 3, 697–708. 10.1515/forum-2020-0354Search in Google Scholar

[11] S. N. Hu, S. F. Hong and W. Zhao, The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015), 135–153. 10.1016/j.jnt.2015.04.006Search in Google Scholar

[12] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, 2nd ed., Grad. Texts in Math. 84, Springer, New York, 1990. 10.1007/978-1-4757-2103-4Search in Google Scholar

[13] S. A. Katre and A. R. Rajwade, Resolution of the sign ambiguity in the determination of the cyclotomic numbers of order 4 and the corresponding Jacobsthal sum, Math. Scand. 60 (1987), no. 1, 52–62. 10.7146/math.scand.a-12171Search in Google Scholar

[14] N. M. Katz, On a theorem of Ax, Amer. J. Math. 93 (1971), 485–499. 10.2307/2373389Search in Google Scholar

[15] R. Lidl and H. Niederreiter, Finite Fields, 2nd ed., Encyclopedia Math. Appl. 20, Cambridge University, Cambridge, 1997. 10.1017/CBO9780511525926Search in Google Scholar

[16] P. J. McCarthy, Introduction to Arithmetical Functions, Universitext, Springer, New York, 1986. 10.1007/978-1-4613-8620-9Search in Google Scholar

[17] O. Moreno and C. J. Moreno, Improvements of the Chevalley–Warning and the Ax–Katz theorems, Amer. J. Math. 117 (1995), no. 1, 241–244. 10.2307/2375042Search in Google Scholar

[18] G. Myerson, On the numbers of zeros of diagonal cubic forms, J. Number Theory 11 (1979), no. 1, 95–99. 10.1016/0022-314X(79)90023-4Search in Google Scholar

[19] G. Myerson, Period polynomials and Gauss sums for finite fields, Acta Arith. 39 (1981), no. 3, 251–264. 10.4064/aa-39-3-251-264Search in Google Scholar

[20] W. M. Schmidt, Equations Over Finite Fields. An Elementary Approach, Lecture Notes in Math. 536, Springer, Berlin, 1976. 10.1007/BFb0080437Search in Google Scholar

[21] T. Storer, Cyclotomy and Difference Sets, Markham, Chicago, 1967. Search in Google Scholar

[22] D. Q. Wan, Zeros of diagonal equations over finite fields, Proc. Amer. Math. Soc. 103 (1988), no. 4, 1049–1052. 10.1090/S0002-9939-1988-0954981-2Search in Google Scholar

[23] D. Q. Wan, An elementary proof of a theorem of Katz, Amer. J. Math. 111 (1989), no. 1, 1–8. 10.2307/2374476Search in Google Scholar

[24] E. Warning, Bemerkung zur vorstehenden Arbeit von Herrn Chevalley, Abh. Math. Semin. Univ. Hambg. 11 (1935), no. 1, 76–83. 10.1007/BF02940715Search in Google Scholar

[25] A. Weil, On some exponential sums, Proc. Natl. Acad. Sci. USA 34 (1948), 204–207. 10.1007/978-1-4757-1705-1_48Search in Google Scholar

[26] J. Wolfmann, The number of solutions of certain diagonal equations over finite fields, J. Number Theory 42 (1992), no. 3, 247–257. 10.1016/0022-314X(92)90091-3Search in Google Scholar

[27] J. Wolfmann, New results on diagonal equations over finite fields from cyclic codes, Finite Fields: Theory, Applications, and Algorithms, Contemp. Math. 168, American Mathematical Society, Providence (1994), 387–395. 10.1090/conm/168/01716Search in Google Scholar

Received: 2021-07-29
Revised: 2021-11-04
Published Online: 2022-01-28
Published in Print: 2022-03-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
https://doi.org/10.1515/forum-2021-0196
Keywords for this article
Diagonal quartic form; cyclotomic number; generating function; finite fields

Articles in the same Issue

  1. Frontmatter
  2. On the variance of the nodal volume of arithmetic random waves
  3. On length densities
  4. On weighted compactness of commutators of bilinear maximal Calderón–Zygmund singular integral operators
  5. Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities
  6. On cutting blocking sets and their codes
  7. Inertia groups and smooth structures on quaternionic projective spaces
  8. On the number of zeros of diagonal quartic forms over finite fields
  9. Weak martingale Hardy-type spaces associated with quasi-Banach function lattice
  10. A dual version of Huppert's ρ-σ conjecture for character codegrees
  11. A topological correspondence between partial actions of groups and inverse semigroup actions
  12. On homotopy braids
  13. Holomorphic convexity of pseudoconvex spaces in terms of the rank of structural sheaf
  14. The Shintani double zeta functions
  15. Homological dimensions relative to preresolving subcategories II
  16. Simplicity of indecomposable set-theoretic solutions of the Yang–Baxter equation
  17. A converse theorem for quasimodular forms
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2026 De Gruyter Brill
Downloaded on 3.2.2026 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2021-0196/pdf
Scroll to top button