Let R(t,w)∈C(t,w) be rational in t and w, and assume that R is not a polynomial in w of degree at most 2. If a function f meromorphic on C satisfies f'(z)=R(e^z,f(z)), must there exist a positive integer q and a rational function S∈C(t) such that f(z)=S(e^z/q) for all z∈C?

Question 6.2 in Gundersen's collection

Let $R(t,w)\in\mathbb{C}(t,w)$ be rational in $t$ and $w$ , and assume that $R$ is not a polynomial in $w$ of degree at most $2$ . If a function $f$ meromorphic on $\mathbb{C}$ satisfies $f'(z)=R(e^z,f(z))$ , must there exist a positive integer $q$ and a rational function $S\in\mathbb{C}(t)$ such that $f(z)=S(e^{z/q})$ for all $z\in\mathbb{C}$ ?

References

Primary source

arXiv

Additional references

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

complex-analysis differential-equations

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A newly reported preprint claims to answer the question affirmatively in a restricted coefficient setting, but the result has not been independently verified.

Gundersen’s Question 6.2 concerns whether meromorphic solutions admit the algebraic form $S(e^{z/q})$ without a growth assumption. The reported result also identifies the algebraic degree with the least deck period.

September 9, 2026 report

A preprint titled Meromorphic solutions of first-order differential equations with rational exponential coefficients is reported to establish the affirmative result. Its scope is limited to rational-exponential coefficients, so it may not settle every formulation or intended generality of the collection’s question.

Current status (as of September 2026): An affirmative answer is claimed for the rational-exponential coefficient setting, but the claim is unverified and broader applicability remains open.

Sources

No solutions have been posted yet.