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arXiv · 2609.23674

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 $\mathbb{P}^n$ 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 $n\geq 3$, 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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Alexandre Eremenko, Zongben Xu, Teng Zhang. 2026-09-20. The modified Cartan conjecture. https://arxiv.org/abs/2609.23674

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