Local height arguments toward the dynamical Mordell-Lang conjecture

@misc{yang2026local,
  title = {{Local height arguments toward the dynamical Mordell-Lang conjecture}},
  author = {She Yang and Aoyang Zheng},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2605.11676},
  eprint = {2605.11676},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.11676},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Local height arguments toward the dynamical Mordell-Lang conjecture
AU  - She Yang
AU  - Aoyang Zheng
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2605.11676
UR  - https://arxiv.org/abs/2605.11676
AB  - We consider regular endomorphisms of the complex affine space with a degree gap k. They are endomorphisms f of 𝔸_ℂ^N of the form f(x_1,…,x_N)=(f_1(x_1,…,x_N)+g_1(x_1,…,x_N),…,f_N(x_1,…,x_N)+g_N(x_1,…,x_N)), in which f_1,…,f_N are homogeneous polynomials of degree d with no nonzero common zeros and g_1,…,g_N are polynomials of degree ≤ d−k. Such an endomorphism extends to an endomorphism of ℙ_ℂ^N. Let H_∞=ℙ_ℂ^N∖𝔸_ℂ^N be the infinity hyperplane and we denote f_∞ as the induced endomorphism of H_∞. Suppose that k is twice greater than the multiplicities of f_∞ at the periodic closed points, i.e. k>2max_P∈Per(f_∞)e_f_∞(P). Then we prove that f satisfies the dynamical Mordell-Lang conjecture for curves. As a by-product of our proof, we show that in this case every periodic curve of f is a "vertical line", i.e. a straight line passing through the origin. There are many examples which satisfy our condition k>2max_P∈Per(f_∞)e_f_∞(P). Indeed, we prove that for every d≥2, a general endomorphism f_∞ of H_∞≅ℙ_ℂ^N−1 of degree d satisfies max_P∈ H_∞(ℂ)e_f_∞(P)≤(N−1)!·2^N−1. So if we take k=(N−1)!·2^N+1, then f will satisfy our condition if f_∞ is general (of an arbitrary degree d≥ k). Moreover, we provide examples to illustrate that this condition is optimal to force every periodic curve to be a vertical line, in the sense that one cannot change ">" into "≥".
ER  -