{
  "id": "2210.05748",
  "version": "v1",
  "published": "2022-10-11T19:27:38.000Z",
  "updated": "2022-10-11T19:27:38.000Z",
  "title": "Critical Points at Infinity for Hyperplanes of Directions",
  "authors": [
    "Stephen Gillen"
  ],
  "categories": [
    "math.CO",
    "cs.SC"
  ],
  "abstract": "Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of the coefficients of a meromorphic generating function $F = G/H$ in a direction $\\mathbf{r}$. It uses Morse theory on the pole variety $V := \\{ H = 0 \\} \\subseteq (\\mathbb{C}^*)^d$ of $F$ to deform the torus $T$ in the multivariate Cauchy Integral Formula via the downward gradient flow for the \\textit{height} function $h = h_{\\mathbf{r}} = -\\sum_{j=1}^d r_j \\log |z_j|$, giving a homology decomposition of $T$ into cycles around \\textit{critical points} of $h$ on $V$. The deformation can flow to infinity at finite height when the height function is not a proper map. This happens only in the presence of a critical point at infinity (CPAI): a sequence of points on $V$ approaching a point at infinity, and such that log-normals to $V$ converge projectively to $\\mathbf{r}$. The CPAI is called \\textit{heighted} if the height function also converges to a finite value. This paper studies whether all CPAI are heighted, and in which directions CPAI can occur. We study these questions by examining sequences converging to faces of a toric compactification defined by a multiple of the Newton polytope $\\mathcal{P}$ of the polynomial $H$. Under generically satisfied conditions, any projective limit of log-normals of a sequence converging to a face $F$ must be parallel to $F$; this implies that CPAI must always be heighted and can only occur in directions parallel to some face of $\\mathcal{P}$. When this generic condition fails, we show under a smoothness condition, that a point in a codimension-1 face $F$ can still only be a CPAI for directions parallel to $F$, and that the directions for a codimension-2 face can be a larger set, which can be computed explicitly and still has positive codimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-11T19:27:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical point",
      "hyperplanes",
      "directions parallel",
      "multivariate cauchy integral formula",
      "height function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}