{
  "abstract": "We consider regular endomorphisms of the complex affine space with a degree gap $k$. They are endomorphisms $f$ of $\\mathbb{A}_{\\mathbb{C}}^{N}$ of the form $f(x_1,\\dots,x_N)=(f_1(x_1,\\dots,x_N)+g_1(x_1,\\dots,x_N),\\dots,f_N(x_1,\\dots,x_N)+g_N(x_1,\\dots,x_N))$, in which $f_1,\\dots,f_N$ are homogeneous polynomials of degree $d$ with no nonzero common zeros and $g_1,\\dots,g_N$ are polynomials of degree $\\leq d-k$. Such an endomorphism extends to an endomorphism of $\\mathbb{P}_{\\mathbb{C}}^{N}$. Let $H_{\\infty}=\\mathbb{P}_{\\mathbb{C}}^{N}\\setminus\\mathbb{A}_{\\mathbb{C}}^{N}$ be the infinity hyperplane and we denote $f_{\\infty}$ as the induced endomorphism of $H_{\\infty}$. Suppose that $k$ is twice greater than the multiplicities of $f_{\\infty}$ at the periodic closed points, i.e. $k>2\\max\\limits_{P\\in\\mathrm{Per}(f_\\infty)}e_{f_{\\infty}}(P)$. 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 $k>2\\max\\limits_{P\\in\\mathrm{Per}(f_\\infty)}e_{f_{\\infty}}(P)$. Indeed, we prove that for every $d\\geq2$, a general endomorphism $f_{\\infty}$ of $H_{\\infty}\\cong\\mathbb{P}_{\\mathbb{C}}^{N-1}$ of degree $d$ satisfies $\\max\\limits_{P\\in H_{\\infty}(\\mathbb{C})}e_{f_{\\infty}}(P)\\leq(N-1)!\\cdot2^{N-1}$. So if we take $k=(N-1)!\\cdot2^N+1$, then $f$ will satisfy our condition if $f_{\\infty}$ is general (of an arbitrary degree $d\\geq k$). 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 \"$\\geq$\".",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.11676",
  "author_total": 2,
  "authors": [
    {
      "name": "She Yang"
    },
    {
      "name": "Aoyang Zheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bt2c12p7h5nxt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.11676",
  "fulltext_url": null,
  "id": "bt2c12p7h5nxt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.11676",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7161045154",
  "pdf_url": "https://arxiv.org/pdf/2605.11676",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bt2c12p7h5nxt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2605.11676",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.11676"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Local height arguments toward the dynamical Mordell-Lang conjecture",
  "title_text": "Local height arguments toward the dynamical Mordell-Lang conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/bt2c12p7h5nxt",
  "venue": "arXiv",
  "versions": []
}