有限域上广义Markoff-Hurwitz-type方程的有理点个数

doi:10.3785/j.issn.1008-9497.2017.05.003

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摘要设N_q表示有限域F_q上广义Markoff-Hurwitz-type方程的有理点个数（a₁x₁^m₁+a₂x₂^m₂+…+a_nx_n^m_n）^k=cx₁^k₁ x₂^k₂…x_t^k_t，其中n ≥ 2， m_i， k， k_j和t ≥ n是正整数，a_i，c属于F_q^*，其中1 ≤ i ≤ n， 1 ≤ j ≤ t. 最近有研究推广了Carlitz的结果，给出了上述方程当k=k₁=…=k_t=1时的有理点个数. 当未定元的指数满足一定条件时，本文给出了上述广义方程的有理点个数，推广了已有结论.

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	胡双年
	李艳艳

关键词 ：有限域, 有理点, Markoff-Hurwitz-type方程

Abstract：Let N_q denote the number of solutions of the generalized Markoff-Hurwitz-type equations (a₁x₁^m₁+a₂x₂^m₂+…+a_nx_n^m_n)^k=cx₁^k₁ x₂^k₂…x_t^k_tover the finite field F_q,where n ≥ 2,m_i,k,k_j and t ≥ n are positive integers,a_i,c∈F_q^*,for 1 ≤ i ≤ n and 1 ≤ j ≤ t.Recently,some researches considered the above equation with k=k₁=…=k_t=1 and obtained some generalizations of Carlitz's results.In this paper,we determine N_q explicitly in some other cases.This extends the previous conclusions.

Key words： finite field rational point Markoff-Hurwitz-type equations

收稿日期: 2016-11-07

ZTFLH:

O156.1

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.

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http://www.zjujournals.com/sci/CN/10.3785/j.issn.1008-9497.2017.05.003 或 http://www.zjujournals.com/sci/CN/Y2017/V44/I5/516

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