{
  "abstract": "Let $n\\ge2$ be even, let $λ=(λ_1,\\ldots,λ_n)\\in\\mathbb{R}^n$ have pairwise distinct coordinates, and define the difference-power matrix \\[ A_d(λ) := \\bigl[(λ_r-λ_s)^d\\bigr]_{r,s=1}^n, \\qquad d\\in\\mathbb{N}. \\] In 1928, Colombo proved that $\\det A_{n-1}(λ)\\ne0$---and hence $\\det A_{n-1}(λ)>0$---and that $\\operatorname{rank} A_d(λ)=d+1$ for $0\\le d<n-1$. He conjectured that \\[ \\det A_d(λ)\\ne0 \\qquad\\text{for every } d\\ge n-1. \\] For even $d$, the conjectured nonsingularity follows from previously published results on distance-power matrices. The remaining open cases were therefore the supercritical odd exponents $d\\ge n+1$. We prove nonsingularity for all these odd exponents, thereby completing Colombo's conjecture. Consequently, \\[ \\operatorname{rank} A_d(λ)=\\min\\{n,d+1\\} \\qquad(d\\in\\mathbb{N}). \\] Our proof converts a hypothetical kernel vector into a real binary form having more projective real linear factors, counted with multiplicity, than its real Waring length permits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28274",
  "author_total": 4,
  "authors": [
    {
      "name": "Kun Li"
    },
    {
      "name": "Li Tie"
    },
    {
      "name": "Peng Wang"
    },
    {
      "name": "Zihan Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aq87w8avr2nb9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28274",
  "fulltext_url": null,
  "id": "aq87w8avr2nb9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28274",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7204443264",
  "pdf_url": "https://arxiv.org/pdf/2608.28274",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aq87w8avr2nb9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28274",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28274"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An algebraic proof of Colombo's difference-power determinant conjecture",
  "title_text": "An algebraic proof of Colombo's difference-power determinant conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/aq87w8avr2nb9",
  "venue": "arXiv",
  "versions": []
}