Rational Solutions of First Order Algebraic Ordinary Differential Equations

2022/01/27 by Shuang Feng, Feng, Shuang, Li-Yong Shen +1
Computer Science · Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Computer and information sciences #FOS: Mathematics #Symbolic Computation (cs.SC) #cs.SC #math.CA

paper · pdf · doi:10.48550/arxiv.2201.11378

arXiv admin note: substantial text overlap with arXiv:2005.01289

arxiv created 2022/01/27 · arxiv updated 2022/01/28

Abstract

Let f(t,y,y')=∑_i=0ⁿ 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, we show that if f satisfies deg(f,y)<deg(f,y') or max_i=0ⁿ \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'.

Citations

• Rational Solutions of First Order Algebraic Ordinary Differential Equations

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