{
  "abstract": "We associate to a collision $p\\ne q$ of a homogeneous Keller map $F=\\operatorname{id}+h$ the polarization subalgebra generated by $p$ and $q$. This collision hull is the smallest invariant linear subspace containing the collision; the restricted map remains a noninjective Keller map, and its dimension controls that of the standard symmetric lift. The construction is compatible with scalar extension and equivariant under linear conjugacy, so it gives a canonical linear carrier for a marked failure of injectivity. We compute this carrier exactly in two recent cubic-homogeneous reductions of the three-variable Jacobian counterexample. For Thompson's 24-variable map, the growth $2,4,11,20,20$ recovers MacFarlane's 20-dimensional invariant subspace. For the 19-variable homogenization of Van Rijn's 12-variable degree-three map, the growth $2,4,11,19,19$ fills the whole space. Hence no invariant linear restriction retaining the displayed collision can improve the corresponding 38-variable symmetric lift. We give the resulting 340-monomial homogeneous quartic explicitly: it is Hessian-nilpotent, violates Zhao's Vanishing Conjecture, and its gradient Keller map has an exact collision over $\\mathbb{Q}(i)$. The analogous 20-variable application gives the previously studied 40-variable, 350-monomial quartic. All finite calculations are checked by exact companion code.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.12543",
  "author_total": 1,
  "authors": [
    {
      "name": "Thomas Prellberg"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8je24a3nsxa12.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.12543",
  "fulltext_url": null,
  "id": "8je24a3nsxa12",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.12543",
  "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"
    ],
    [
      "openalex",
      "2610|Mathematical Dynamics and Fractals"
    ]
  ],
  "openalex": "W7203448814",
  "pdf_url": "https://arxiv.org/pdf/2608.12543",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8je24a3nsxa12.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2608.12543",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.12543"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Collision-Generated Compression for Homogeneous Keller Maps",
  "title_text": "Collision-Generated Compression for Homogeneous Keller Maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/8je24a3nsxa12",
  "venue": "arXiv",
  "versions": []
}