Meromorphic solutions of first-order differential equations with rational exponential coefficients

RESUMO

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.