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

  • Junyong Zhao , Yulu Feng , Shaofang Hong EMAIL logo und Chaoxi Zhu
VerĂśffentlicht/Copyright: 28. Januar 2022
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 34 Heft 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.

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

Š 2022 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  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
SchlagwĂśrter fĂźr diesen Artikel
Diagonal quartic form; cyclotomic number; generating function; finite fields

Artikel in diesem Heft

  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
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