Meromorphic solutions of algebraic differential equations

First-order algebraic differential equations of genus zero

“…Remark (i)Goldberg proved that the solutions to have finite order of growth actually, . (ii)Eremenko considered the same class of equations . Among others, he gave a proof of Malmquist's Second Theorem ( the existence of some solutions with essential singularity at implies the validity of the Fuchsian conditions ) in the classical and also the more general setting of ‘admissible’ solutions.…”

“…In the most simple case q = 1, equation (1) reduces to a Riccati equation (3). Based on the properties of the solutions to (3) we will prove the following theorem, which gives a comprehensive description of the solutions with essential singularity at ∞ to equations (1) of genus zero.…”

“….]'. Apparently, the same seems to be perfectly valid also for[3]. One reason might be that its style is too 'algebraic' for classical analysts.…”

“…Many authors seemed to be unaware of Malmquist's Second Theorem, or were in doubt about Malmquist's reasoning. One reason might be that several elegant and transparent proofs for his First Theorem were known, but none for the second one; see also the comments in Eremenko [, p. 62] and footnote 2 in this paper. A priori it is, however, not the same to ask for (necessary and sufficient) conditions for the existence of one transcendental meromorphic solution on one hand, and for the absence of movable singularities except poles on the other.…”

“…† In[3] the author wrote: 'The paper [42] (Malmquist's paper[10] here) has had practically no influence on the work of other authors [. .…”