Research Problems in Function Theory — Problem 1.37

Definitions and mathematical statement

For a nonzero meromorphic function $f$ on $\mathbb C$, the order $\operatorname{ord}_a(f)$ is the integer $m$ in a local expression $f(z)=(z-a)^m u(z)$ with $u$ holomorphic and $u(a)\ne0$; positive $m$ indicates a zero and negative $m$ a pole. The two legible requirements in the source are \[ \operatorname{ord}_a(f)>0\ \Longrightarrow\ \operatorname{ord}_a(f)\in2\mathbb Z_{>0}, \] \[ \operatorname{ord}_a(f)<0\ \Longrightarrow\ -\operatorname{ord}_a(f)\in\{1,3,5,\ldots\} \qquad(a\in\mathbb C). \] Problem (J. Winkler). Find criteria for, and/or give explicit methods for constructing, meromorphic functions on $\mathbb C$ with these zero- and pole-multiplicity properties. The source requires explicit methods to have all computations practicable, not merely an abstract existence proof.

The extraction has broken list syntax, including an isolated (a) and empty semicolon entries. The display above treats the two surviving requirements as simultaneous, but does not invent any lost condition. No prescribed divisor, growth bound, nonconstancy requirement, or required number of zeros or poles is stated; functions with no zeros or no poles satisfy the corresponding universal condition vacuously.

The source gives the background question of meromorphic solutions in the whole plane to \[ y''+A(z)y=0, \] but specifies neither the coefficient class for $A$ nor the relation between $f$ and such solutions. This background is retained without adding an unprovided differential-equation constraint to $f$.

Short English statement

Find practical criteria or constructions for plane meromorphic functions whose zeros all have even multiplicity and whose poles all have odd multiplicity. The damaged list and the unspecified differential-equation connection limit the precision of the task.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 1.37, attributed to J. Winkler, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The 2018 update was retained for status triage.
• Local registry data/unsolvedmath/problems.json, permanent ID $2301037$, source code catalogue entry, Research Problems in Function Theory --- Problem 1.37. The record labels this open and OPEN-TRIAGE and reports no recent resolution located, with NEEDS_REVIEW source-status and rights qualifications. No separate resolution or background-paper citation is supplied.