Critical Points at Infinity for Hyperplanes of Directions

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.