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Holomorphic curves of finite lower order with few inflection points
Abstract
We prove a conjecture proposed by the first-named author in 1998. Let f:ℂ→ℙn be a transcendental linearly non-degenerate holomorphic curve of finite lower order. If the counting function N1(r,f) of its Wronskian zeros satisfies N1(r,f)=o(T(r,f)), then its order and lower order coincide and belong to {1+k/q:k∈ℤ≥ 0, 2≤ q≤ n+1}, and its characteristic is regularly varying. Every order in this set occurs. We also prove the sharp inequality limsupr→∞N1(r,f)/T(r,f)≥ 1 for transcendental linearly non-degenerate curves of order zero.
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BibTeX
@misc{eremenko2026holomorphic,
title = {{Holomorphic curves of finite lower order with few inflection points}},
author = {Alexandre Eremenko and Teng Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38032},
eprint = {2609.38032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38032},
}
RIS
TY - UNPB
TI - Holomorphic curves of finite lower order with few inflection points
AU - Alexandre Eremenko
AU - Teng Zhang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38032
UR - https://arxiv.org/abs/2609.38032
AB - We prove a conjecture proposed by the first-named author in 1998. Let f:ℂ→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∈ℤ_≥ 0, 2≤ q≤ n+1}, and its characteristic is regularly varying. Every order in this set occurs. We also prove the sharp inequality limsup_r→∞N_1(r,f)/T(r,f)≥ 1 for transcendental linearly non-degenerate curves of order zero.
ER -