{
  "abstract": "We study parametrized systems of $\\mathit{generalized}$ polynomial inequalities, $(c \\circ x^B) \\in C$, for a positive variable vector $x \\in \\mathbb{R}^n_>$, a $\\mathit{real}$ exponent matrix $B \\in \\mathbb{R}^{n \\times m}$, a positive parameter vector $c \\in \\mathbb{R}^m_>$, and a positive, blunt coefficient cone $C \\subset \\mathbb{R}^m_>$. This setting includes (i) generalized polynomial equations, $A \\, (c \\circ x^B) = 0$, arising e.g. from power-law dynamical systems with real exponents (and involving a real coefficient matrix $A \\in \\mathbb{R}^{n' \\times m}$), and (ii) generalized semi-algebraic sets. Our main result guarantees the generic finiteness of solutions modulo unavoidable exponential fibers. Building on a framework for parametrized $\\mathit{polynomial}$ systems developed by Müller and Regensburger, we reformulate the problem as a system of $\\mathit{binomial}$ equations on the coefficient set $P = C \\cap Δ\\subseteq \\mathbb{R}^m_>$, where $Δ$ is a product of (standard) simplices. Solvability is governed by the monomial-dependency subspace $D$. Using elementary arguments from analysis and topology, including a logarithmic boundary analysis and Sard's theorem, we control the accumulation of solutions on the boundary and in the interior of $P$. We show that, if $\\dim P \\le \\dim D$, then generically in both the measure-theoretic and topological sense, there are finitely many solutions on $P$ (and hence finitely many exponential fibers of solutions in $\\mathbb{R}^n_>$). As an application, we prove generic finiteness of steady states (within forward-invariant sets) for parametrized power-law dynamical systems with real exponents.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28588",
  "author_total": 1,
  "authors": [
    {
      "name": "Stefan Müller"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6mntfcb0a8fj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28588",
  "fulltext_url": null,
  "id": "c6mntfcb0a8fj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28588",
  "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"
    ],
    [
      "arxiv",
      "q-bio.MN"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7223104437",
  "pdf_url": "https://arxiv.org/pdf/2610.28588",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6mntfcb0a8fj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28588",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28588"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.MN",
      "subfield": "systems-synthetic"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Generic finiteness of solutions to parametrized systems of generalized polynomial inequalities",
  "title_text": "Generic finiteness of solutions to parametrized systems of generalized polynomial inequalities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/c6mntfcb0a8fj",
  "venue": "arXiv",
  "versions": []
}