{
  "id": "2208.04590",
  "version": "v1",
  "published": "2022-08-09T08:11:26.000Z",
  "updated": "2022-08-09T08:11:26.000Z",
  "title": "New bounds for the number of connected components of fewnomial hypersurfaces",
  "authors": [
    "Bihan Frédéric",
    "Humbert Tristan",
    "Tavenas Sébastien"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "We prove that the zero set of a \\(5\\)-nomial in \\(n\\) variables, whose exponent vectors are not colinear, has at most $\\lfloor \\frac{n-1}{2}\\rfloor +3$ connected components in the positive orthant. Moreover, we give an explicit \\(5\\)-nomial in $2$ variables which defines a curve with three connected components in the posititive orthant, showing that our bound is sharp for $n=2$. In a more general setting, if \\(\\A = \\{a_0,\\ldots,a_{d+k}\\} \\subset \\Z^n\\) has dimension \\(d\\), we prove that the number of connected components of the zero set of a \\(\\A\\)-polynomial in the positive orthant is smaller than or equal to \\(18 (d+1)^{k-1} 2^{\\binom{k-1}{2}} + k+1\\), improving the previously known bounds. Moreover, our results continue to work for polynomials with real exponents.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-09T08:11:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connected components",
      "fewnomial hypersurfaces",
      "zero set",
      "positive orthant",
      "exponent vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}