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Session S24 - Symbolic Computation: Theory, Algorithms and Applications

Tuesday, July 20, 16:30 ~ 16:55 UTC-3

A New Approach to the Connection Problem Using Analytic Combinatorics in Several Variables

Stephen Melczer

University of Waterloo, Canada   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

A foundational problem in enumerative and analytic combinatorics asks one to take an integer sequence defined by some kind of recurrence and determine its dominant asymptotic behaviour. For sequences satisfying linear recurrences with polynomial coefficients, an asymptotic decomposition can often be deduced as a $\mathbb{C}$-linear combination of a finite set of explicit functions. Unfortunately, there is currently no known algorithm to determine these "connection" coefficients, or even decide which are non-zero. In this talk we describe a new approach to compute connection coefficients for sequences arising as "diagonals" of multivariate rational functions. Our approach combines new Morse-theoretic tools from the study of analytic combinatorics in several variables (ACSV) with computer algebra methods for rigorous numeric analytic continuation of functions satisfying linear ODEs. We illustrate this approach on our motivating application, a series of positivity conjectures of Straub and Zudilin.

Joint work with Yuliy Baryshnikov (University of Illinois, Urbana-Champaign) and Robin Pemantle (University of Pennsylvania).

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