Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract)

TL;DR

This paper introduces a new systematic method for deriving asymptotics of coefficients in multivariate algebraic generating functions by embedding them into rational series and applying analytic combinatorics in several variables (ACSV) theory.

Contribution

It proposes a novel embedding approach that simplifies asymptotic analysis of algebraic series by leveraging rational series and ACSV theory, offering a promising alternative to direct algebraic methods.

Findings

Systematic asymptotic results for multivariate algebraic generating functions.

Embedding algebraic series into rational series facilitates analysis.

Potential for further improvements with refined embedding techniques.

Abstract

We present a strategy for computing asymptotics of coefficients of $d$-variate algebraic generating functions. Using known constructions, we embed the coefficient array into an array represented by a rational generating functions in $d+1$ variables, and then apply ACSV theory to analyse the latter. This method allows us to give systematic results in the multivariate case, seems more promising than trying to derive analogs of the rational ACSV theory for algebraic GFs, and gives the prospect of further improvements as embedding methods are studied in more detail.