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arXiv · 2608.26062

Three omitted values and non-Blaschke point divisors in half-planes

Quanyu Tang·Bokai Cui·Wei He·Tao Hu·Yanyang Li·Ke Wang·Zijun Yu
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Abstract

We construct a real meromorphic function $F$ on $\mathbb C$ such that $F^{-1}(\{0,1,\infty\})\subset\mathbb R$, while $F$ is not of bounded type in either half-plane. More strongly, for every $a\in\widehat{\mathbb C}\setminus\{0,1,\infty\}$, the $a$-point divisor in either half-plane fails the Blaschke condition. Thus the construction provides an independent negative answer to a question going back to Nevanlinna's 1925 work that had remained open for over a century. Postcomposition gives the analogous counterexample for any prescribed triple of distinct values in the Riemann sphere. The core construction and proof were generated during an autonomous run of GPT-5.6 Sol Ultra.

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Quanyu Tang, Bokai Cui, Wei He, Tao Hu, Yanyang Li, Ke Wang, Zijun Yu. 2026-08-26. Three omitted values and non-Blaschke point divisors in half-planes. https://arxiv.org/abs/2608.26062

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