Critical Points at Infinity for Hyperplanes of Directions

TL;DR

This paper investigates the nature of critical points at infinity in the context of analytic combinatorics in several variables, showing that such points are always heighted and only occur in specific directions related to the Newton polytope faces.

Contribution

It provides a detailed analysis of when and how critical points at infinity occur, establishing conditions under which they are heighted and characterizing their directional limitations.

Findings

CPAI are always heighted under generic conditions.

CPAI can only occur in directions parallel to faces of the Newton polytope.

Explicit computation of possible CPAI directions for codimension-2 faces.

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.