{
  "abstract": "Consider a system of equations $A(x)\\cdot F(x) = B(x)$ on a subset $X$ of $\\mathbb R^n$, where $A(x)$ and $B(x)$ are matrix- and vector-valued semialgebraic functions on $X$, and the unknown $F(x)$ is a field of vector-valued $m$-jets. We assume there is a solution which is the field of Taylor polynomials of order $m$ on $X$ of a $\\mathcal C^m$ vector-valued function $f$ on $\\mathbb R^n$, and ask whether we can find a $\\mathcal C^m$ semialgebraic solution $f$. Our main result is a positive answer in the case $\\dim X = 1$. The methods are based on an article of Fefferman and Luli for $X \\subset \\mathbb R^2$, and a secondary goal is to show that their approach applies in a much simpler way in the case $\\dim X =1$, or in the case of the semialgebraic Whitney extension problem in $\\mathbb R^2$. Our results hold more generally for functions definable in an o-minimal expansion of $\\mathbb R$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07210",
  "author_total": 2,
  "authors": [
    {
      "name": "Edward Bierstone"
    },
    {
      "name": "Jean-Baptiste Campesato"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0x2f5647tg1dd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07210",
  "fulltext_url": null,
  "id": "0x2f5647tg1dd",
  "issns": [],
  "landing_url": "https://hal.science/hal-05782645v1",
  "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.AG"
    ],
    [
      "arxiv",
      "math.CA"
    ],
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      "arxiv",
      "math.CV"
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      "math.LO"
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      "2610|Mathematical and Theoretical Analysis"
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  "openalex": "W7220939198",
  "pdf_url": "https://arxiv.org/pdf/2610.07210",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0x2f5647tg1dd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:50Z",
      "native_id": "hal-05782645",
      "source": "hal",
      "url": "https://hal.science/hal-05782645v1"
    },
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07210",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07210"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "$\\mathcal C^m$ solutions of semialgebraic equations on curves",
  "title_text": "𝒞^m solutions of semialgebraic equations on curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/0x2f5647tg1dd",
  "venue": "HAL",
  "versions": []
}