The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields
HU Shuangnian1,2, LI Yanyan3
1. School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, Henan Province, China;
2. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China;
3. School of Electronic and Electrical Engineering, Nanyang Institute of Technology, Nanyang 473004, Henan Province, China
Abstract:Let Nq denote the number of solutions of the generalized Markoff-Hurwitz-type equations (a1x1m1+a2x2m2+…+anxnmn)k=cx1k1x2k2…xtktover the finite field Fq,where n ≥ 2,mi,k,kj and t ≥ n are positive integers,ai,c∈Fq*,for 1 ≤ i ≤ n and 1 ≤ j ≤ t.Recently,some researches considered the above equation with k=k1=…=kt=1 and obtained some generalizations of Carlitz's results.In this paper,we determine Nq explicitly in some other cases.This extends the previous conclusions.
基金资助:Supported by the Key Program of Universities of Henan Province of China (17A110010),China Postdoctoral Science Foundation Funded Project (2016M602251) and by the National Science Foundation of China Grant (11501387).
作者简介: 胡双年(1982-),ORCID:http://orcid.org/0000-0002-5174-8460,male,Ph.D,lecturer,the field of interest is number theory,E-mail:hushuangnian@163.com.
引用本文:
胡双年, 李艳艳. 有限域上广义Markoff-Hurwitz-type方程的有理点个数[J]. 浙江大学学报(理学版), 2017, 44(5): 516-519,537.
HU Shuangnian, LI Yanyan. The number of solutions of generalized Markoff-Hurwitz-type equations over finite fields. Journal of ZheJIang University(Science Edition), 2017, 44(5): 516-519,537.
[1] MARKOFF A A. Sur les formes quadratiques binaires indéfinies[J]. Mathematische Annalen,1880,17(3):379-399.
[2] HURWITZ A. Über eine aufgabe der unbestimmten analysis[J]. Archiv der Mathematik und Physik,1907(3):185-196.
[3] CARLITZ L. Certain special equations in a finite field[J]. Monatshefte Für Mathematik,1954,58(1):5-12.
[4] CARLITZ L. The number of solutions of some equations in a finite field[J]. Portugaliae Mathematica,1954,13(1):25-31.
[5] BAOULINA I. On the number of solutions of the equation a1x1m1+a2x2m2+…+anxnmn=bx1x2…xn in a finite field[J]. Acta Applicandae Mathematicae,2005,89(1):35-39.
[6] BAOULINA I. Generalizations of the Markoff-Hurwitz equations over finite fields[J]. Journal of Number Theory,2006,118(1):31-52.
[7] ÖZDEMIR M E, YILDIZ Ç, AKDEMIR A O, et al. On some inequalities for s-convex functions and applications[J]. Journal of Inequalities and Applications,2013,2013:333.
[8] SARIKAYA M Z, SET E, YALDIZ H, et al. Hermite-Hadamard's inequalities for fractional integrals and related fractional inequalities[J]. Math Comput Model,2013,57:2403-2407.
[9] SET E. New inequalities of Ostrowski type for mappings whose derivatives are s-convex in the second sense via fractional integrals[J]. Comput Math Appl,2012,63:1147-1154.
[10] WANG J R, ZHU C C, ZHOU Y. New generalized Hermite-Hadamard type inequalities and applications to special means[J]. Journal of Inequalities and Applications,2013,2013:325.
[11] CAO W, SUN Q. On a class of equations with special degrees over finite fields[J].Acta Arithmetica,2007,130(2):195-202.
[12] ZHAO Z J, CAO X W. On the number of solutions of certain equations over finite fields[J].Journal of Mathematical Research and Exposition,2010,30(6):957-966.
[13] KENG H L.Introduction to Number Theory[M]. Heidelberg:Springer-Verlag,1982.
[14] BAOULINA I. Solutions of equations over finite fields:Enumeration via bijections[J].Journal of Algebra and Its Applications,2016,15(7):1650136.
[15] LIDL R, NIEDERREITER H.Finite Fields-Encyclopedia of Mathematics and Its Applications[M]. 2nd ed. Cambridge:Cambridge University Press,1997.