{
  "abstract": "We give a unified account of zero counting in finite-field polynomial rings with commuting variables twisted by a common automorphism. A decomposition into fixed-field slices connects skew evaluation with ordinary polynomial evaluation. First, we derive a slice coefficient formula from the skew Combinatorial Nullstellensatz and use it to deduce skew Chevalley-Warning, including the sufficient condition $n>h\\sum_i d_i$. Next, classical Ax-Katz gives the corrected integer congruence $Q^{\\max\\{0,\\lceil(n-h\\sum_i d_i)/D\\rceil\\}}\\mid N+(M-1)δ$. We construct homogeneous examples proving optimality for every positive degree table and every number of variables. We also establish a sharp lower bound with a prescribed zero, an input-sensitive coefficient-filtration refinement, and transfers of known bounds for nonlinear zero sets. Detailed calculations explain the origin correction, incomplete norm blocks, and the distinction between characteristic-$p$ summation and higher integer divisibility. Existing geometric, interpolation, and counting theorems are explicitly attributed.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35739",
  "author_total": 3,
  "authors": [
    {
      "name": "Shuang Li"
    },
    {
      "name": "Xuan Pang"
    },
    {
      "name": "Pingzhi Yuan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ftyyvw8sbc0k9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35739",
  "fulltext_url": null,
  "id": "ftyyvw8sbc0k9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35739",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7223998804",
  "pdf_url": "https://arxiv.org/pdf/2610.35739",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ftyyvw8sbc0k9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.35739",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35739"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "From skew Combinatorial Nullstellensatz to sharp Ax-Katz congruences and zero counts",
  "title_text": "From skew Combinatorial Nullstellensatz to sharp Ax-Katz congruences and zero counts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/ftyyvw8sbc0k9",
  "venue": "arXiv",
  "versions": []
}