Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.38032

Holomorphic curves of finite lower order with few inflection points

Alexandre Eremenko·Teng Zhang
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

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

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ross-Witt Nyström correspondence and Ohsawa-Takegoshi extension

This paper is an attempt to study a few \emph{effective} universality phenomenons in complex geometry using deformations and the Hörmander $L^2$-method. The main idea of the proof is to deform the general case to the toric (or convex) case and to apply two new Berndtsson-Lempert type Ohsawa-Takegoshi extension theorems. The first one is based on the Ross-Witt Nyström correspondence picture, Darvas-Xia-Zhang's asymptotic slope formula for Ding-type functionals and Berndtsson's monotonicity theorem. The second version is proved using an $S^1$-symmetrization method for PSH potentials based on Witt Nyström's canonical Kähler deformations and Berndtsson-P\u aun's positivity theorem for Stein fibrations associated to deformations to normal bundles. The resulting effective universality phenomenons include a sharp lower bound of the Bergman kernel for compact Riemann surfaces, an effective Okounkov body construction and a sharp Faber-Widom type Bergman approximation of the logarithmic capacity.

math.CV↗
arXiv2026

Regularity of solutions to the complex Monge--Ampère equation with a general modulus of continuity

We investigate the regularity theory for the complex Monge--Ampère equation when, owing to the data or to the underlying domain, its solution fails to be in any Hölder class and to have a general concave modulus of continuity. Our first main result is motivated by, and generalizes, a result of S.-Y. Li. As an application, we prove a result on the regularity of solutions to the complex Monge--Ampère equation on general $B$-regular domains in $\mathbb{C}^n$.

math.CV↗
arXiv2026

Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis

This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the $L^2$-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman--Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of $ζ$ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping $e^{-k\eps^2}$ is discussed as an open problem.

math.CV↗