{
  "id": "1905.13282",
  "version": "v1",
  "published": "2019-05-30T20:04:51.000Z",
  "updated": "2019-05-30T20:04:51.000Z",
  "title": "Two remarks on sums of squares with rational coefficients",
  "authors": [
    "Jose Capco",
    "Claus Scheiderer"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "There exist homogeneous polynomials $f$ with $\\mathbb Q$-coefficients that are sums of squares over $\\mathbb R$ but not over $\\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields $K/\\mathbb Q$ with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any $f$ as above necessarily has a (non-trivial) real zero. In the minimal open cases $(3,6)$ and $(4,4)$, we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-30T20:04:51.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational coefficients",
      "ingredient totally imaginary number fields",
      "real zero",
      "specific galois-theoretic properties",
      "minimal open cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}