Given an autonomous first order algebraic ordinary differential equation $F(y,y^{\prime })=0$, we prove that every formal Puiseux series solution of $F(y,y^{\prime })=0$, expanded around any finite point or at infinity, is convergent. The proof is constructive and we provide an algorithm to describe all such Puiseux series solutions. Moreover, we show that for any point in the complex plane there exists a solution of the differential equation which defines an analytic curve passing through this point.

中文翻译：

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自治一阶微分方程Puiseux级数解的存在性和收敛性

给定自治的一阶代数常微分方程 $F（ÿ，{ÿ}^{\prime }）=0$，我们证明了Puiseux系列的所有正式解决方案 $F（ÿ，{ÿ}^{\prime }）=0$围绕任意有限点或无限远展开的，是收敛的。该证明具有建设性，我们提供了一种算法来描述所有此类Puiseux级数解。而且，我们表明，对于复平面中的任何点，都存在微分方程的解，该解定义了一条穿过该点的解析曲线。