高秩径向分歧斜积的动力 Mordell--Lang 猜想

中文摘要 AI 辅助

我们在复数域上证明了任意基维数和纤维秩的径向分歧多项式斜积族的动力 Mordell--Lang 猜想。我们假设某一次固定迭代将竖直临界轨迹送入不变零截面。我们的结果覆盖仿射线性基，并在度差条件下覆盖非线性平展多项式基。我们还为固定的非线性轨道和固定多项式系数线性递推建立了统一的 Skolem--Mahler--Lech 定理，其中递推的尾系数为非零常数。该定理适用于涉及该轨道和任意解的有限多个连续项的联合多项式与有理关系。尽管有限例外集可能依赖于关系，但单个最终周期适用于每个这样的解和关系。我们的证明将受控非阿基米德实现与超吸引线丛动力学的轨道传递定理相结合。

英文摘要

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.

发表机构

• The University of Memphis（孟菲斯大学）

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