{
  "abstract": "For a smooth complete intersection $V\\subset(\\C^*)^n$ of dimension $k$, $n\\ge2k$, we bound the volume of its amoeba by $π^{2k}$ times a sum of mixed volumes, divided by the order of any finite group $Γ$ of symmetries commuting with the logarithmic map. For $n=2k\\ge4$ almost every fibre of the amoeba map has a number of points divisible by $2|Γ|$. For $n=2k$ we characterise the varieties of maximal volume, show that for $k\\ge2$ they are not real, and prove that no plane in general position in $(\\C^*)^4$ has maximal volume. For $n>2k$ the bound is never attained, varieties of maximal volume have amoeba maps injective off a null set, and injectivity is generic. The amoeba map of almost every real $k$-plane is two-to-one, and that of the $k$-th symmetric power of a real line exactly $2^k$-to-one; conversely, real $k$-planes in general position in $(\\C^*)^{2k}$ with $2^k$-to-one amoeba maps are symmetric powers. We also classify the real planes of $(\\C^*)^4$ and study lines in $(\\C^*)^3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02496",
  "author_total": 1,
  "authors": [
    {
      "name": "Mounir Nisse"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9tv721vmz4e05.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02496",
  "fulltext_url": null,
  "id": "9tv721vmz4e05",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02496",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02496",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9tv721vmz4e05.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02496",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02496"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "The topology of the amoeba map and maximal volume",
  "title_text": "The topology of the amoeba map and maximal volume",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/9tv721vmz4e05",
  "venue": "arXiv",
  "versions": []
}