Advertisement

Results in Mathematics

, 74:37 | Cite as

Counting Rational Points of an Algebraic Variety over Finite Fields

  • Shuangnian Hu
  • Xiaoer Qin
  • Junyong ZhaoEmail author
Article
  • 9 Downloads

Abstract

Let \({\mathbb {F}}_q\) denote the finite field of odd characteristic p with q elements (\(q=p^{n},n\in {\mathbb {N}} \)) and \({\mathbb {F}}_q^*\) represent the nonzero elements of \({\mathbb {F}}_{q}\). In this paper, by using the Smith normal form we give a formula for the number of rational points of the algebraic varieties defined by the following system of equations over \({\mathbb {F}}_{q}\):
$$\begin{aligned} \left\{ \begin{array}{ll} \sum \limits _{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} \ldots x_{n_1}^{e^{(1)}_{i,n_1}} +\sum \limits _{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} \ldots x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ \sum \limits _{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} \ldots x_{n_3}^{e^{(2)}_{j,n_3}} +\sum \limits _{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} \ldots x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right. \end{aligned}$$
where the integers \(1\le r_1<r_2\), \(1\le r_3<r_4\), \(1\le n_1<n_2\), \(1\le n_3<n_4\), \(n_1\le n_3\), \(b_1, b_2\in \mathbb {F}_{q}\), \(a_{1i}\in \mathbb {F}_{q}^{*}(1\le i\le r_2)\), \(a_{2j}\in \mathbb {F}_{q}^{*}(1\le j\le r_4)\) and the exponent of each variable is positive integer. Our result provides a partial answer to an open problem raised in Hu et al. (J Number Theory 156:135–153, 2015).

Keywords

Finite field algebraic variety rational point Smith normal form exponent matrix 

Mathematics Subject Classification

11T06 11T71 

Notes

References

  1. 1.
    Adolphson, A., Sperber, S.: \(p\)-Adic estimates for exponential sums and the theorem of Chevalley–Warning. Ann. Sci. Ècole Norm. Sup. 20, 545–556 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Adolphson, A., Sperber, S.: \(p\)-Adic Estimates for Exponential Sums, \(p\)-Adic Analysis (Trento, 1989). Lecture Notes in Mathematics, vol. 1454, pp. 11–22. Springer, Berlin (1990)zbMATHGoogle Scholar
  3. 3.
    Ax, J.: Zeros of polynomials over finite fields. Am. J. Math. 86, 255–261 (1964)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Cao, W., Sun, Q.: On a class of equations with special degrees over finite fields. Acta Arith. 130, 195–202 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Carlitz, L.: Pairs of quadratic equations in a finite field. Am. J. Math. 76, 137–154 (1954)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cohen, E.: Congruence representations in algebraic number field. Trans. Am. Math. Soc. 75, 444–470 (1953)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hodges, J.H.: Representations by bilinear forms in a finite field. Duke Math. J. 22, 497–509 (1955)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Hong, S.F.: Newton polygons of L-functions associtated with exponential sums of polynomials of degree four over finite fields. Finite Fields Appl. 7, 205–237 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hong, S.F.: Newton polygons of L-functions associtated with exponential sums of polynomials of degree six over finite fields. J. Number Theory 97, 368–396 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hong, S.F.: L-functions of twisted diagonal exponential sums over finite fields. Proc. Am. Soc. 135, 3099–3108 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hong, S.F., Zhao, J.R., Zhao, W.: The universal Kummer congruences. J. Aust. Math. Soc. 94, 106–132 (2013)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hu, S.N., Zhao, J.Y.: The number of rational points of a family of algebraic varieties over finite fields. Algebra Colloq. 24, 705–720 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hu, S.N., Hong, S.F., Zhao, W.: The number of rational points of a family of hypersurfaces over finite fields. J. Number Theory 156, 135–153 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hua, L.-K.: Introduction to Number Theory. Springer, Berlin (1982)Google Scholar
  15. 15.
    Hua, L.-K., Vandiver, H.S.: Characters over certain types of rings with applications to the theory of equations in a finite field. Proc. Nat. Acad. Sci. 35, 94–99 (1949)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Huang, H., Gao, W., Cao, W.: Remarks on the number of rational points on a class of hypersurfaces over finite fields. Algebra Colloq. 25, 533–540 (2018)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, GTM 84, 2nd edn. Springer, New York (1990)CrossRefGoogle Scholar
  18. 18.
    Katz, N.M.: On a theorem of Ax. Am. J. Math. 93, 485–499 (1971)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and Its Applications, vol. 20, 2nd edn. Cambridge University Press, Cambridge (1997)Google Scholar
  20. 20.
    Moreno, O., Moreno, C.J.: Improvement of Chevalley–Warning and the Ax–Katz theorem. Am. J. Math. 117, 241–244 (1995)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Smith, H.J.S.: On systems of linear indeterminate equations and congruences. Philos. Trans. R. Soc. Lond. 151, 293–326 (1861)CrossRefGoogle Scholar
  22. 22.
    Song, J., Chen, Y.: The number of solutions of some special systems of equations over finite fields. Sci. Sin. Math. 46, 1815–1828 (2016). (in Chinese) CrossRefGoogle Scholar
  23. 23.
    Sun, Q.: On diagonal equations over finite fields. Finite Fields Appl. 3, 175–179 (1997)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sun, Q.: On the formula of the number of solutions of some equations over finite fields. Chin. Ann. Math. 18A, 403–408 (1997). (in Chinese) zbMATHGoogle Scholar
  25. 25.
    Wan, D.: An elementary proof of a theorem of Katz. Am. J. Math. 111, 1–8 (1989)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Wan, D.: Modular counting of rational points over finite fields. Found. Comput. Math. 8, 597–605 (2008)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Wang, W., Sun, Q.: The number of solutions of certain equations over a finite field. Finite Fields Appl. 11, 182–192 (2005)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Wang, W., Sun, Q.: An explicit formula of solution of some special equations over a finite field. Chin. Ann. Math. 26A, 391–396 (2005). (in Chinese) MathSciNetzbMATHGoogle Scholar
  29. 29.
    Wolfmann, J.: The number of solutions of certain diagonal equations over finite fields. J. Number Theory 42, 247–257 (1992)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wolfmann, J.: New Results on Diagonal Equations Over Finite Fields from Cyclic Codes. Finite Fields: Theory, Applications, and Algorithms (Las Vegas, NV, 1993). Contemporary Mathematics, vol. 168, pp. 387–395. American Mathematical Society, Providence (1994)zbMATHGoogle Scholar
  31. 31.
    Yang, J.: A class of systems of equations over a finite field. Acad. Forum Nan Du (Nat. Sci. Ed.) 20, 7–12 (2000). (in Chinese) Google Scholar
  32. 32.
    Yang, J.M.: An explicit formula for the number of solutions of a certain system of equations over a finite field. Acta Math. Sin. (Chin. Ser.) 50, 653–660 (2007)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNanyang Institute of TechnologyNanyangPeople’s Republic of China
  2. 2.College of Mathematics and StatisticsYangtze Normal UniversityChongqingPeople’s Republic of China
  3. 3.Mathematical CollegeSichuan UniversityChengduPeople’s Republic of China

Personalised recommendations