Tutte Colloquium - Stephen Gillen

Title: Critical Points at Infinity for Hyperplanes of Directions

Abstract: Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of series coefficients of multivariate rational functions G/H in an exponent direction r. The poly-torus of integration T that arises from the multivariate Cauchy Integral Formula is deformed away from the origin into cycles around critical points of a “height function" h on V = V(H). The deformation can sometimes flow to infinity at finite height in the presence of a critical point at infinity (CPAI): a sequence of points on V approaching a point at infinity p, and such that the log-normals to V converge projectively to the direction of r. The CPAI is called heighted if the height function also converges to a finite value. In this talk we discuss under what conditions we know that all CPAI are heighted, and in which directions CPAI can occur, by compactifying in a toric variety X_A. In smooth cases under generically satisfied conditions, CPAI must always be heighted. When this generic condition fails but X_A is sufficiently smooth at p, it is often possible to show that a point in a codimension-1 face F can still only be a CPAI for directions parallel to F, but that the directions for a codimension-2 face can be a larger set, which can be computed explicitly and still has positive codimension.