Homotopy techniques for analytic combinatorics in several variables

TL;DR

This paper introduces a practical algorithm combining homotopy continuation and analytic combinatorics for deriving asymptotics of multivariate rational generating functions, applicable beyond non-negativity assumptions.

Contribution

It presents the first implementation that computes asymptotics for multivariate rational functions without relying on non-negativity, using homotopy methods in a computational setting.

Findings

Successfully terminates on 3-variable examples from literature.

Heuristic methods predict asymptotics in higher dimensions.

Implemented in Julia with HomotopyContinuation.jl, demonstrating practical effectiveness.

Abstract

We combine tools from homotopy continuation solvers with the methods of analytic combinatorics in several variables to give the first practical algorithm and implementation for the asymptotics of multivariate rational generating functions not relying on a non-algorithmically checkable `combinatorial' non-negativity assumption. Our homotopy implementation terminates on examples from the literature in three variables, and we additionally describe heuristic methods that terminate and correctly predict asymptotic behaviour in reasonable time on examples in even higher dimension. Our results are implemented in Julia, through the use of the HomotopyContinuation.jl package, and we provide a selection of examples and benchmarks.