{
  "abstract": "We study the growth of the bidegree of an ample irreducible curve in $\\mathbb{P}^1 \\times \\mathbb{P}^1$ under a product polynomial endomorphism $φ=(f,g)$, where at least one of $f$ and $g$ is non-exceptional. We prove that, if the curve $C$ is not preperiodic under $(f^a,g^b)$ for any $a,b\\geq 1$, then the bidegree of $φ^n(C)$ is asymptotic to $(\\text{deg}(g)^n,\\text{deg}(f)^n)$. As an application of this exponential growth, we prove a geometric analogue of Silverman's theorem on the finiteness of $S$-integral points in orbits. Namely if $C$ is not $(f^a,g^b)$-preperiodic and $C'$ is not totally invariant for $φ$, then for any infinite sequence ${n_i}$ of positive integers, the union of the intersections $$ \\bigcup_{i \\geq 1} \\left( φ^{n_i}(C)\\cap C' \\right) $$ is Zariski dense in $C'$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20580",
  "author_total": 2,
  "authors": [
    {
      "name": "Sina Saleh"
    },
    {
      "name": "Jit Wu Yap"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8pbb8dnzyqma5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20580",
  "fulltext_url": null,
  "id": "8pbb8dnzyqma5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20580",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213585810",
  "pdf_url": "https://arxiv.org/pdf/2609.20580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8pbb8dnzyqma5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.20580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Degree Growth of Iterates of Curves and Likely Intersections",
  "title_text": "Degree Growth of Iterates of Curves and Likely Intersections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/8pbb8dnzyqma5",
  "venue": "arXiv",
  "versions": []
}