Rational Solutions of First Order Algebraic Ordinary Differential Equations  

作　　者：FENG Shuang SHEN Liyong 

机构地区：[1]School of Physical and Mathematical Sciences,Nanjing Tech University,Nanjing 211816,China [2]School of Mathematical Sciences,University of Chinese Academy of Sciences,Beijing 100049,China

出　　处：《Journal of Systems Science & Complexity》2024年第2期567-580,共14页系统科学与复杂性学报（英文版）

基　　金：supported by Beijing Natural Science Foundation under Grant No.Z190004;the National Key Research and Development Project under Grant No.2020YFA0713703;the Fundamental Research Funds for the Central Universities.

摘　　要：Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y' but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)<deg(f,y')or n max i=0{deg(a_(i),y)−2(n−i)}>0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.

关 键 词：Degree bound first order AODE HEIGHT rational solution 

分 类 号：O175.1[理学—数学]