{
  "abstract": "There is a natural connection between the SOS rank problem for bi-quadratic forms and the Zarankiewicz extremal problem for C4-free bipartite graphs. The classical Zarankiewicz number z(m,n) controls the bipartite skeleton associated with monomial squares. Allowing two cells to form a single bilinear square leads to augmented Zarankiewicz configurations and the second-order Zarankiewicz number z2(m,n). Unlike zRL and zSL, defined through recursive sufficient conditions, z2 maximizes over all irreducible displayed SOS decompositions without imposing (RW 3+). Hence, to prove z2(m,n)<=R, one must prove that every simple limited configuration with more displayed squares is reducible; failure of a sufficient condition cannot serve as a counterargument. We prove z2(4,4)=10, z2(7,4)=19, z2(8,4)=21, z2(5,5)=17, and obtain z2=zSL=zRL in all these cases. The extremal irreducible 6x4 configurations with 16 displayed squares form a single isomorphism class under row and column relabeling, whereas the extremal irreducible 7x4 configurations with 19 displayed squares form exactly three isomorphism classes. The four-column results form a structural chain: classify lower-order extremal configurations first, then use hereditary irreducibility under deletion of complete squares to constrain the next order. Finite exhaustive steps use candidate pruning, a necessary compatibility graph, clique enumeration, and orbit reduction, with a verifiable reducibility or irreducibility proof for each remaining orbit. For 5x5, there are two ordinary extremal skeletons; finite exclusion leaves only two highly symmetric 18-square candidates. They define the same ten-square polynomial, which admits an explicit nine-square representation, yielding the upper bound for z2(5,5).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18429",
  "author_total": 2,
  "authors": [
    {
      "name": "Yi Xu"
    },
    {
      "name": "Xihong Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cv6d3kmmywnjh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18429",
  "fulltext_url": null,
  "id": "cv6d3kmmywnjh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18429",
  "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.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7213525687",
  "pdf_url": "https://arxiv.org/pdf/2609.18429",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cv6d3kmmywnjh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18429",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18429"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact Values, Extremal Classifications, and Sum-of-Squares Reductions for Second-Order Zarankiewicz Numbers",
  "title_text": "Exact Values, Extremal Classifications, and Sum-of-Squares Reductions for Second-Order Zarankiewicz Numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/cv6d3kmmywnjh",
  "venue": "arXiv",
  "versions": []
}