2020/05/04 by Ruyong Feng, Feng, Ruyong, Shuang Feng +1 · 1 citation
Computer Science · Mathematics · #34A05 #68W30 #Advanced Differential Equations and Dynamical Systems #Algebraic number #Combinatorics #Constant (computer programming) #Degree (music) #Differential equation #Discrete mathematics #FOS: Computer and information sciences #Mathematical analysis #Mathematics #Numerical methods for differential equations #Order (exchange) #Ordinary differential equation #Physics #Polynomial #Polynomial and algebraic computation #Symbolic Computation (cs.SC) #Upper and lower bounds #cs.SC #msc:34A05 #msc:68W30
paper · pdf · doi:10.48550/arxiv.2005.01289
40 pages
arxiv created 2020/05/04 · openalex publication_date 2020/05/04 · arxiv updated 2020/05/05 · openalex created_date 2024/01/26 · openalex updated_date 2026/07/28
Let f(t, y,y')=∑i=0d ai(t, y)y'i=0 be a first order ordinary differential equation with polynomial coefficients. Eremenko in 1999 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 polynomial in t,y,y'. In this paper, we show that if maxi=0d \\rm deg(ai,y)-2(d-i)\>0 then the degree bound C only depends on the degrees of f, and furthermore we present an explicit expression for C in terms of the degrees of f.