发表机构
Institute of Applied Mathematics, Shenzhen Polytechnic University; School of Mathematics and Information Science, Guangzhou University; School of Basic and General Education, Guangzhou University of Software（深圳职业技术大学应用数学研究所; 广州大学数学与信息科学学院; 广州软件学院基础与通识教育学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究一阶微分方程亚纯解，证明除非系数为二次多项式，否则解必为代数形式，并给出Gundersen问题6.2的肯定回答，通过几何方法恢复Riccati方程。
AI 中文摘要
我们研究一阶微分方程 $f'=R(e^z,f)$，其中 $R\in\C(t,w)$。我们证明，在整个复平面上的亚纯解在 $\C(e^z)$ 上是代数的，除非 $R$ 在其第二个变量中是次数至多为2的多项式。这样的代数解必然具有形式 $S(e^{z/q})$，其中 $S$ 是有理的，$q$ 是正整数，从而对 Gundersen 文集中问题6.2给出了肯定回答。几何输入是 Guillot 关于曲面上亚纯向量场的单值轨道的定理。额外的论证将该定理提供的纤维化与原始指数坐标进行比较。一个分歧计算迫使每个有限非零分支值 $a$ 满足 $Da=a$，这排除了这样的值，并将比较简化为两点覆盖。因子整除性和相对代数闭包随后在原始系数域上恢复了一个 Riccati 方程。没有施加增长性假设。我们还确定了代数次数与最小整数覆盖周期，并描述了所得解的增长性和值纤维。
英文摘要
We study first-order differential equations $f'=R(e^z,f)$, where $R\in\C(t,w)$. We prove that a meromorphic solution on the whole complex plane is algebraic over $\C(e^z)$ unless $R$ is a polynomial of degree at most two in its second variable. Such an algebraic solution necessarily has the form $S(e^{z/q})$, with $S$ rational and $q$ a positive integer, giving an affirmative answer to Question~6.2 in Gundersen's collection. The geometric input is Guillot's theorem on single-valued trajectories of meromorphic vector fields on surfaces. The additional argument compares the fibration provided by that theorem with the original exponential coordinate. A ramification calculation forces every finite nonzero branch value $a$ to satisfy $Da=a$, which excludes such a value and reduces the comparison to a two-point cover. Divisor divisibility and relative algebraic closedness then recover a Riccati equation over the original coefficient field. No growth hypothesis is imposed. We also identify the algebraic degree with the least integer deck period and describe the growth and value fibres of the resulting solutions.
