{
  "title": "A coefficient normalization for positive definite forms",
  "upload_type": "publication",
  "publication_type": "preprint",
  "publication_date": "2026-09-23",
  "creators": [
    {
      "name": "Kriebel, Alec",
      "affiliation": "Independent researcher",
      "orcid": "0009-0001-9320-500X"
    }
  ],
  "description": "<p>Every positive definite real homogeneous form of positive degree admits an invertible real linear change of variables in which the symmetric ordered coefficients have diagonal entries equal to one and every mixed entry strictly between zero and one. This gives an affirmative answer to Brandes's Problem 11 in the 2019 Oberwolfach report on Analytic Number Theory, including the printed absolute-value inequality.</p><p>The three-page proof clusters a basis near a direction minimizing the form on a Euclidean sphere. A polynomial gap has a strictly positive quadratic term for every mixed tuple. The accompanying source package includes an independent proof via positive definite bilinear slices and Cauchy-Schwarz, internal adversarial audits including three fresh preprint-readiness rounds and independently rechecked supplementary controls, a dated priority search, and a standard-library Python verifier with exact rational certificates. The finite checks supplement the analytic proof and are not a formal verification of the universal theorem.</p><p>Brandes's 2015 discussion of pseudo-diagonal forms is credited as a direct precursor. A bounded literature audit found no earlier full resolution; first priority is not established. This is an unrefereed preprint developed and checked with generative-AI assistance. Paper and prose: CC BY 4.0. Code: MIT.</p>",
  "access_right": "open",
  "license": "cc-by-4.0",
  "version": "1.0",
  "language": "eng",
  "keywords": [
    "positive definite forms",
    "homogeneous polynomials",
    "symmetric polarization",
    "coefficient normalization",
    "pseudo-diagonal forms",
    "Brandes problem",
    "OWR-17293-016"
  ],
  "related_identifiers": [
    {
      "identifier": "10.4171/OWR/2019/50",
      "relation": "references",
      "scheme": "doi"
    },
    {
      "identifier": "10.1112/jlms/jdv028",
      "relation": "references",
      "scheme": "doi"
    },
    {
      "identifier": "https://github.com/AlecKriebel/Math/tree/main/owr_17293_016_brandes_normalization",
      "relation": "isSupplementedBy",
      "scheme": "url"
    },
    {
      "identifier": "https://aleckriebel.github.io/Math/papers/brandes-coefficient-normalization/",
      "relation": "isDocumentedBy",
      "scheme": "url"
    }
  ]
}