二阶代数微分方程组的超越亚纯解  

Transcendental meromorphic solutions of systems of second-order algebraic differential equations

作　　者：任国珍 高凌云[1] REN Guozhen;GAO Lingyun(Department of Mathematics,Jinan University,Guangzhou 510632,China)

机构地区：[1]暨南大学数学系,广州510632

出　　处：《成都理工大学学报（自然科学版）》2020年第2期249-256,共8页Journal of Chengdu University of Technology: Science & Technology Edition

基　　金：国家自然科学基金项目(11271161)。

摘　　要：利用亚纯函数的Nevanlinna值分布理论和最大模原理,讨论了如下二阶代数微分方程组存在超越亚纯解时,该解的多项式的零点问题,{(w″1)n=P1(z,w2)/Q1(z,w2) (w″2)n=P2(z,w1)/Q2(z,w1)得到2个结果。结果表明如果该类二阶代数微分方程组存在超越亚纯解(w1(z),w2(z))时,且V1(z,w2)与V2(z,w1)分别为Q1(z,w2)与Q2(z,w1)或P1(z,w2)与P2(z,w1)的质因子,则V1(z)=V1(z,w2)与V2(z)=V2(z,w1)中至少有一个具有无穷多个零点。该结果推广了一阶复微分方程的结论。Based on the distribution of Nevanlinna value of meromorphic functions and maximum modulus principle,the zero problems of polynomials is discussed if the following second-order algebraic differential equations exist beyond the meromorphic solutions.{(w″1)n=P1(z,w2)/Q1(z,w2) (w″2)n=P2(z,w1)/Q2(z,w1)The obtained two results show if the second-order algebraic differential equations of this class have prime factors beyond meromorphic solutions(w1(z),w2(z)),and V1(z,w2)and V2(z,w1)are Q1(z,w2)and Q2(z,w1)or P1(z,w2)and P2(z,w1),respectively,then V1(z)=V1(z,w2),and V2(z)=V2(z,w1),at least one of them has an infinite number of zeros.The presented results generalize the conclusion of the first-order complex differential equations.

关 键 词：超越亚纯函数 微分方程组 最大模原理 

分 类 号：O174.52[理学—数学]