论文标题
一阶代数普通微分方程的合理解决方案
Rational Solutions of First Order Algebraic Ordinary Differential Equations
论文作者
论文摘要
令$ f（t，y，y'）= \ sum_ {i = 0}^n a_i（t，y）y'^i = 0 $是带有多项式系数的不可约的一阶普通微分方程。 1998年的Eremenko证明了存在一个恒定的$ c $，因此每个合理解决方案$ f（t，y，y'）= 0 $的程度不大于$ c $。示例表明，该学位限制的$ c $不仅取决于$ t，y，y'$中的$ f $的学位，而且还取决于$ f $ of $ t，y，y'$的$ f $的系数。在本文中，我们表明，如果$ f $满足$ deg（f，y）<deg（f，y'）$或$ \ max_ {i = 0}^n \ {deg（a_i，y）-2（a_i，y）-2（n-i）\}> 0 $，那么$ c $ bung的$ c $仅取决于$ f $ $ f $ $ f $ $ t，y y y y y y y y y'$'$'$，y y'$'$'$'' $ t，y，y'$的$ f $的学位。
Let $f(t,y,y')=\sum_{i=0}^n a_i(t,y)y'^i=0$ 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, we show that if $f$ satisfies $deg(f,y)<deg(f,y')$ or $\max_{i=0}^n \{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 $t,y,y'$.
