{
  "abstract": "The moment-sum-of-squares (SOS) hierarchy provides a sequence of bounds on the minimum of a polynomial over a basic closed semialgebraic set, i.e., a subset of $\\mathbb{R}^n$ defined by finitely many polynomial constraints. We establish optimal bounds on the error of the SOS hierarchy in the following settings: 1. If the constraints define an Archimedean quadratic module: the error of the Putinar-based hierarchy at level $k$ is $Θ((\\log k)^4/k^2)$ in the worst case. 2. If the constraints define a compact semialgebraic set: the error of the Schmüdgen-based hierarchy at level $k$ is $Θ((\\log k)^2/k^2)$ in the worst case. 3. If the constraints define an Archimedean quadratic module and satisfy linear independence constraint qualification conditions (LICQ): the error of the hierarchy at level $k$ is $Θ(1/k^{4})$ in the worst case. Here, the upper bound holds already for the Putinar-based hierarchy, while the lower bound holds even against the Schmüdgen-based hierarchy. Equivalently, these give best-possible effective versions of Putinar's and Schmüdgen's Positivstellensätze for these settings (in terms of their dependence on $k$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24638",
  "author_total": 9,
  "authors": [
    {
      "name": "Alexander Taveira Blomenhofer"
    },
    {
      "name": "Sander Gribling"
    },
    {
      "name": "Didier Henrion"
    },
    {
      "name": "Milan Korda"
    },
    {
      "name": "Poornendu Kumar"
    },
    {
      "name": "Victor Magron"
    },
    {
      "name": "Lucas Slot"
    },
    {
      "name": "Luis Felipe Vargas"
    },
    {
      "name": "Juan Vera"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1a41r9rjqhh3j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24638",
  "fulltext_url": null,
  "id": "1a41r9rjqhh3j",
  "issns": [],
  "landing_url": "https://laas.hal.science/hal-05825413v1",
  "language": "en",
  "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.OC"
    ],
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-oc"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7223025519",
  "pdf_url": "https://arxiv.org/pdf/2610.24638",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1a41r9rjqhh3j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:59Z",
      "native_id": "hal-05825413",
      "source": "hal",
      "url": "https://laas.hal.science/hal-05825413v1"
    },
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24638",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24638"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Optimal degree bounds for sum-of-squares representations",
  "title_text": "Optimal degree bounds for sum-of-squares representations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/1a41r9rjqhh3j",
  "venue": "HAL",
  "versions": []
}