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arXiv · 2609.38032

Holomorphic curves of finite lower order with few inflection points

Alexandre Eremenko·Teng Zhang
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Abstract

We prove a conjecture proposed by the first-named author in 1998. Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function $N_1(r,f)$ of its Wronskian zeros satisfies $N_1(r,f)=o(T(r,f))$, then its order and lower order coincide and belong to $\{1+k/q:k\in\mathbb{Z}_{\geq 0},\ 2\le q\le n+1\}$, and its characteristic is regularly varying. Every order in this set occurs. We also prove the sharp inequality $\limsup_{r\to\infty}N_1(r,f)/T(r,f)\geq 1$ for transcendental linearly non-degenerate curves of order zero.

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Alexandre Eremenko, Teng Zhang. 2026-09-29. Holomorphic curves of finite lower order with few inflection points. https://arxiv.org/abs/2609.38032

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