{
  "abstract": "Let $p$ be a prime. We provide examples which show that étale endomorphisms of affine planes over an algebraically closed field $k$ of characteristic $p$ can have fibers of arbitrary finite cardinal. Let $(l,m)\\in\\mathbb N\\times\\mathbb N^{\\ast}$. We provide examples of such étale endomorphisms whose images have complements of cardinality $l$ and whose geometric degrees are $pm$. Several conjectures are disproved, and in particular we provide an analog over $k$ of Kulikov's counterexample to the `Generalized Jacobian Conjecture' of Bass for surfaces. Surjective (resp.\\ surjective and non-surjective) counterexamples to Adjamagbo's Jacobian Conjecture over each $k$ are included in dimension $2$ (resp.\\ in all dimensions at least $3$); for instance, if $p=2$, then we show for each $m$ there exist surjective étale endomorphisms of the affine spaces over $k$ of dimension at least $3$ of geometric degree $m$. If $e:X\\rightarrow X$ is an endomorphism of a variety over an algebraically closed field $K$, then we show that there exists $n\\in\\mathbb N$ such that $\\Imm(e^n)=\\Imm(e^{n+1})$ provided either (i) $e$ is quasi-finite or (ii) $\\dim(X)\\le 2$ and $X\\setminus\\Imm(e)$ is finite. We provide examples of endomorphisms of affine planes of dimensions at least $3$ over $K$ whose images have complements of cardinality $l$ and for all $n\\in\\mathbb N$ we have $\\Imm(e^n)\\neq\\Imm(e^{n+1})$. We prove that all affine moduli schemes of étale endomorphisms of affine spaces with Jacobian matrices of determinant 1 over $K$ are connected and classify all those that are smooth. We prove that the Jacobian Conjecture over $\\mathbb C$ and Adjamagbo's analog of it over $k$ hold for étale endomorphisms of affine spaces that are composites $g\\circ f$, where $f$ is quasi-finite and a locally closed embedding in codimension $1$ outside a specific finite subset and $g$ is a projection that omits one coordinate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05746",
  "author_total": 3,
  "authors": [
    {
      "name": "Alexander Borisov"
    },
    {
      "name": "Ofer Gabber"
    },
    {
      "name": "Adrian Vasiu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c57wn93mj09ey.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05746",
  "fulltext_url": null,
  "id": "c57wn93mj09ey",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05746",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7211987480",
  "pdf_url": "https://arxiv.org/pdf/2609.05746",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c57wn93mj09ey.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05746",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05746"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On endomorphisms of affine spaces and the Jacobian problem",
  "title_text": "On endomorphisms of affine spaces and the Jacobian problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/c57wn93mj09ey",
  "venue": "arXiv",
  "versions": []
}