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arXiv · 2606.24553

Bialgebraic geometry of B\"ottcher coordinates

Sina Saleh
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Abstract

Becker and Bergweiler showed that if $f$ is a non-exceptional polynomial, then the B\"ottcher coordinate $\Psi_f \colon \mathbb D_R \to B_\infty(f)$ associated to $f$ is a transcendental function. In this paper, we study $f$-bialgebraic sets: algebraic subsets of $\mathbb D_R^n$ whose image under the coordinate-wise action of $\Psi_f$ is contained in an algebraic set of the same dimension. We give a complete dynamical classification of bialgebraic sets under the additional assumption that the Julia set of $f$ is either disconnected, or connected and admits a nondegenerate locally connected model. Inspired by the Ax--Lindemann--Weierstrass theorem and the Ax--Schanuel conjecture, we formulate analogs with $\Psi_f$ in place of the exponential function and prove them in the case where the Julia set $J_f$ is disconnected.

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Sina Saleh. 2026-06-23. Bialgebraic geometry of B\"ottcher coordinates. https://arxiv.org/abs/2606.24553

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