{
  "id": "1803.00539",
  "version": "v1",
  "published": "2018-03-01T18:10:18.000Z",
  "updated": "2018-03-01T18:10:18.000Z",
  "title": "Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare",
  "authors": [
    "Saugata Basu",
    "Antonio Lerario",
    "Abhiram Natarajan"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Given a sequence $\\{Z_d\\}_{d\\in \\mathbb{N}}$ of smooth and compact hypersurfaces in $\\mathbb{R}^{n-1}$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $\\Gamma\\subset \\mathbb{R}\\mathrm{P}^n$ such that each manifold $Z_d$ appears as a component of the zero set on $\\Gamma$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $\\Gamma$ is algebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $\\Gamma$ is bounded by a polynomial in $\\mathrm{deg}(p)$.) We call these \"pathological examples\". In particular, we show that for every $0 \\leq k \\leq n-2$ and every sequence of natural numbers $a=\\{a_d\\}_{d\\in \\mathbb{N}}$ there is a regular, compact and definable hypersurface $\\Gamma\\subset \\mathbb{R}\\mathrm{P}^n$, a subsequence $\\{a_{d_m}\\}_{m\\in \\mathbb{N}}$ and homogeneous polynomials $\\{p_{m}\\}_{m\\in \\mathbb{N}}$ of degree $\\mathrm{deg}(p_m)=d_m$ such that: \\begin{equation} \\label{eq:pathintro} b_k(\\Gamma\\cap Z(p_m))\\geq a_{d_m}.\\end{equation} (Here $b_k$ denotes the $k$-th Betti number.) This generalizes a result of Gwo\\'zdziewicz, Kurdyka and Parusi\\'nski. On the other hand, for a given definable $\\Gamma$ we show that the Fubini-Study measure, in the gaussian space of polynomials of degree $d$, of the set $\\Sigma_{d_m,a, \\Gamma}$ of polynomials verifying $b_k(\\Gamma\\cap Z(p_m))\\geq a_{d_m}$ is positive, but there exists a contant $c_\\Gamma$ such that this measure can be bounded by: \\begin{equation} 0<\\mathbb{P}(\\Sigma_{d_m, a, \\Gamma})\\leq \\frac{c_{\\Gamma} d_m^{\\frac{n-1}{2}}}{a_{d_m}}. \\end{equation} This shows that the set of \"pathological examples\" has \"small\" measure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-03-01T18:10:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "polynomial",
      "pathologies",
      "zero set",
      "pathological examples",
      "th betti number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}