{
  "abstract": "Let $(K^h,v^h)$ be a Henselization of a valued field $(K,v)$, $μ$ any extension of $v$ to $K[x]$, and $μ^h$ the canonical extension of $μ$ and $v^h$ to $K^h[x].$ In this paper, we study two new invariants associated with a Mac Lane-Vaquié chain (MLV) of $μ$, namely, the defect of each augmentation step and the regularity of an MLV chain of $μ$. This, in turn, extends the notion of regularity of a complete sequence of abstract key polynomials for $μ$ to the regularity of $μ$ itself. We also prove that the regularity of $μ$ is a sufficient condition for the equality of the depths of $μ$ and $μ^h$. Furthermore, it is a necessary and sufficient condition for the equality of the defect and relative gap of each augmentation step of an MLV chain of $μ.$ Finally, we show that if $μ$ has finite degree, then the regularity of $μ$ is equivalent to $ \\operatorname{deg}(μ)=\\operatorname{deg}(μ^h)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23443",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikita Dwivedi"
    },
    {
      "name": "Anuj Bishnoi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/11cmt8rarn891.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23443",
  "fulltext_url": null,
  "id": "11cmt8rarn891",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23443",
  "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.AC"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214006641",
  "pdf_url": "https://arxiv.org/pdf/2609.23443",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/11cmt8rarn891.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23443",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23443"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On Regular Mac Lane-Vaquié chains",
  "title_text": "On Regular Mac Lane-Vaquié chains",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/11cmt8rarn891",
  "venue": "arXiv",
  "versions": []
}