Geometry of Gradient Flows for Analytic Combinatorics - Stephen Gillen

Title: Geometry of Gradient Flows for Analytic Combinatorics

Abstract: Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of series coefficients of multivariate rational functions in an exponent direction r by analyzing the singular set V of a multivariate rational function. The poly-torus of integration T that arises from the multivariate Cauchy Integral Formula (it would be a circle in one complex variable) is deformed away from the origin into cycles around critical points of a “height function" h on V. 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, 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. In smooth cases under generically satisfied conditions, CPAI must always be heighted. Non-generic cases are also studied under other conditions.