PreprintarXiv licencearXiv
The Gross Property for Implicit Functions
Abstract
In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. Eremenko asked whether the same conclusion holds for an implicit function defined by an entire relation in two variables. We prove that there exist an entire function F of two variables and a regular implicit germ φ, defined by F(z,φ(z))=0, whose analytic continuation fails on every ray from its centre. In fact, along each ray, the modulus of the continuation tends to infinity as the singular point is approached.
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BibTeX
@misc{nadi2026gross,
title = {{The Gross Property for Implicit Functions}},
author = {Sina Nadi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37451},
eprint = {2609.37451},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37451},
}
RIS
TY - UNPB
TI - The Gross Property for Implicit Functions
AU - Sina Nadi
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.37451
UR - https://arxiv.org/abs/2609.37451
AB - In 1918, Gross proved that a regular local inverse of a meromorphic function in the plane can be continued analytically along every ray from its centre except for directions in a set of Lebesgue measure zero. Eremenko asked whether the same conclusion holds for an implicit function defined by an entire relation in two variables. We prove that there exist an entire function F of two variables and a regular implicit germ φ, defined by F(z,φ(z))=0, whose analytic continuation fails on every ray from its centre. In fact, along each ray, the modulus of the continuation tends to infinity as the singular point is approached.
ER -