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}^d a_i（t，y）y'^i = 0 $是带有多项式系数的第一阶普通微分方程。 1999年的Eremenko证明，存在一个恒定的$ c $，因此每个合理解决方案$ f（t，y，y'）= 0 $的程度不大于$ c $。示例表明，此度限制的$ c $不仅取决于$ t，y，y'$中的$ f $的学位，而且还取决于$ f $的系数，在$ t，y，y'$中被视为多项式。在本文中，我们表明，如果$$ \ max_ {i = 0}^d \ {{{\ rm deg}（a_i，y）-2（d-i）-2（d-i）\}> 0 $$，那么bound的$ c $仅取决于$ f $的$ f $，并且此外，我们还显示了$ c $ c $ c $ c $ n $ n $ f $ n $ f $ f $ f $ f $ f $ f。
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 $f$.
