Rational Solutions of First Order Algebraic Ordinary Differential Equations
| Autor: | Feng, Ruyong, Feng, Shuang |
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| Rok vydání: | 2020 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $f(t, y,y')=\sum_{i=0}^d a_i(t, y)y'^i=0$ 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 $$\max_{i=0}^d \{{\rm deg}(a_i,y)-2(d-i)\}>0 $$ 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 $f$. Comment: 40 pages |
| Databáze: | arXiv |
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