Let f colonC→P^n be a transcendental linearly non-degenerate holomorphic curve with finite lower order, and let N_1(r,f) denote the counting function of the zeros of its Wronskian. If N_1(r,f)=o(T(r,f)) as r→∞, then the order and lower order of f coincide, their common value is of the form 1+k/q for some k∈Z_≥ 0 and q∈2,…,n+1, and the characteristic T(r,f) is regularly varying.

Eremenko’s 1998 conjecture on holomorphic curves with few inflection points

Let $f\colon\mathbb{C}\to\mathbb{P}^n$ be a transcendental linearly non-degenerate holomorphic curve with finite lower order, and let $N_1(r,f)$ denote the counting function of the zeros of its Wronskian. If $N_1(r,f)=o(T(r,f))$ as $r\to\infty$ , then the order and lower order of $f$ coincide, their common value is of the form $1+k/q$ for some $k\in\mathbb{Z}_{\geq 0}$ and $q\in\{2,\ldots,n+1\}$ , and the characteristic $T(r,f)$ is regularly varying.

References

Primary source

arXiv

Additional references

Holomorphic curves of finite lower order with few inflection points — arXiv — Alexandre Eremenko, Teng Zhang

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A 2026 unrefereed preprint claims to settle the conjecture, but nobody independent has checked the proof.

Eremenko’s 1998 conjecture predicts that a holomorphic curve with finite lower order and negligible averaged inflection-point count has rational lower order and a corresponding asymptotic limit. The conjecture is recorded in Eremenko’s 2015 note.

Known results

The $n=1$ case was already known (Eremenko, 2015).
In the special case $N_1(r)\equiv 0$ , Frei’s theorem gives polynomial differential-equation coefficients when the lower order is finite (recorded by Eremenko, 2015).

2026 preprint

Alexandre Eremenko and Teng Zhang claim to prove the proposed order classification under $N_1(r)=o(T(r))$ , together with a sharp inequality in order-zero cases. This is a claimed resolution, but the manuscript is unrefereed and no independent mathematical assessment was found.

Current status (as of September 2026): The conjecture is claimed solved by Eremenko and Zhang’s preprint, but the full argument remains unverified; the cited special cases are established.

Sources

No solutions have been posted yet.