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The modified Cartan conjecture

  • Alexandre Eremenko
  • Zongben Xu
  • Teng Zhang
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Abstract

In 1928 H. Cartan stated a conjecture about holomorphic curves in ℙn parametrized by the unit disc, omitting n+2 hyperplanes in general position. He proved it for n=2. In 1996 the first-named author constructed counterexamples for all n≥ 3, and proposed a modified form of Cartan's conjecture which he proved for n=3. In this paper we prove this modified conjecture in all dimensions. As a byproduct we prove a quantitative version of the classical theorem that analytic functions are linearly dependent if and only if their Wronskian determinant is zero. We interpret our results in terms of the Kobayashi--Royden pseudometric on the complement of n+2 hyperplanes, and on smooth algebraic varieties in the algebraic torus.

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Cite

BibTeXRIS
BibTeX
@misc{eremenko2026modified,
  title = {{The modified Cartan conjecture}},
  author = {Alexandre Eremenko and Zongben Xu and Teng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23674},
  eprint = {2609.23674},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23674},
}
RIS
TY  - UNPB
TI  - The modified Cartan conjecture
AU  - Alexandre Eremenko
AU  - Zongben Xu
AU  - Teng Zhang
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23674
UR  - https://arxiv.org/abs/2609.23674
AB  - In 1928 H. Cartan stated a conjecture about holomorphic curves in ℙ^n parametrized by the unit disc, omitting n+2 hyperplanes in general position. He proved it for n=2. In 1996 the first-named author constructed counterexamples for all n≥ 3, and proposed a modified form of Cartan's conjecture which he proved for n=3. In this paper we prove this modified conjecture in all dimensions. As a byproduct we prove a quantitative version of the classical theorem that analytic functions are linearly dependent if and only if their Wronskian determinant is zero. We interpret our results in terms of the Kobayashi--Royden pseudometric on the complement of n+2 hyperplanes, and on smooth algebraic varieties in the algebraic torus.
ER  -