Dynamical Mordell--Lang Conjecture for Higher-Rank Radially Ramified Skew Products

Abstract

We establish the dynamical Mordell--Lang conjecture over the complex numbers for a family of radially ramified polynomial skew products in arbitrary base dimension and fiber rank. We assume that one fixed iterate sends the vertical critical locus into the invariant zero section. Our result covers affine-linear bases and, under a degree-gap condition, nonlinear étale polynomial bases. We also establish a uniform Skolem--Mahler--Lech theorem for a fixed nonlinear orbit and a fixed polynomial-coefficient linear recurrence whose trailing coefficient is a nonzero constant. The theorem applies to joint polynomial and rational relations involving the orbit and finitely many successive terms of any solution. A single eventual period works for every such solution and relation, although the finite exceptional set may depend on the relation. Our proof combines controlled nonarchimedean realizations with an orbitwise transfer theorem for superattracting line-bundle dynamics.

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BibTeX RIS

BibTeX

@misc{mishra2026dynamical,
  title = {{Dynamical Mordell--Lang Conjecture for Higher-Rank Radially Ramified Skew Products}},
  author = {Bhawesh Mishra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15956},
  eprint = {2609.15956},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15956},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - Dynamical Mordell--Lang Conjecture for Higher-Rank Radially Ramified Skew Products
AU  - Bhawesh Mishra
PY  - 2026
DA  - 2026/09/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.15956
UR  - https://arxiv.org/abs/2609.15956
AB  - We establish the dynamical Mordell--Lang conjecture over the complex numbers for a family of radially ramified polynomial skew products in arbitrary base dimension and fiber rank. We assume that one fixed iterate sends the vertical critical locus into the invariant zero section. Our result covers affine-linear bases and, under a degree-gap condition, nonlinear étale polynomial bases. We also establish a uniform Skolem--Mahler--Lech theorem for a fixed nonlinear orbit and a fixed polynomial-coefficient linear recurrence whose trailing coefficient is a nonzero constant. The theorem applies to joint polynomial and rational relations involving the orbit and finitely many successive terms of any solution. A single eventual period works for every such solution and relation, although the finite exceptional set may depend on the relation. Our proof combines controlled nonarchimedean realizations with an orbitwise transfer theorem for superattracting line-bundle dynamics.
ER  -

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