In this paper we deal with Abel equations of the form $dy/dx=A_1\left(x\right)y+A_2\left(x\right)y^2+A_3\left(x\right)y^3$ , where $A_1\left(x\right),A_2\left(x\right)$ and $A_3\left(x\right)$ are real polynomials and $A_3\neg \equiv 0$ . We prove that these Abel equations can have at most two rational (non-polynomial) limit cycles when $A_1\neg \equiv 0$ and three rational (non-polynomial) limit cycles when $A_1\equiv 0$ . Moreover, we show that these upper bounds are sharp. We show that the general Abel equations can always be reduced to this one.

中文翻译：

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Abel多项式方程的有理极限环

在本文中，我们处理以下形式的Abel方程 $dy/dx=A_1\left(x\right)y+A_2\left(x\right)y^2+A_3\left(x\right)y^3$ ，在哪里 $A_1\left(x\right),A_2\left(x\right)$ 和 $A_3\left(x\right)$ 是实多项式 $A_3\neg \equiv 0$ 。我们证明了这些Abel方程在以下情况下最多可以具有两个有理（非多项式）极限环 $A_1\neg \equiv 0$ 和三个有理（非多项式）极限环 $A_1\equiv 0$ 。此外，我们表明这些上限是尖锐的。我们表明，一般的Abel方程总是可以简化为这一方程。