Critical Points at Infinity for Hyperplanes of Directions
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Analytic combinatorics in several variables (ACSV) analyzes the asymptotic growth of the coefficients of a meromorphic generating function F = G/H in a direction 𝐫. It uses Morse theory on the pole variety V := { H = 0 }⊆ (ℂ^*)^d of F to deform the torus T in the multivariate Cauchy Integral Formula via the downward gradient flow for the height function h = h_𝐫 = -∑_j=1^d r_j log |z_j|, giving a homology decomposition of T into cycles around 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 𝐫. The CPAI is called 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 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 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.
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