A SageMath Package for Analytic Combinatorics in Several Variables: Beyond the Smooth Case
Benjamin Hackl, Andrew Luo, Stephen Melczer, Éric Schost
TL;DR
ACSV studies the asymptotics of multivariate sequences from analytic properties of generating functions. This work tackles the problem for rational $F(\mathbf{z})=\frac{G(\mathbf{z})}{H(\mathbf{z})}$ where the singular variety $\mathcal{V}(H)$ may be non-smooth, by integrating Whitney stratification with ACSV techniques to compute rigorously the diagonals in direction $\mathbf{r}$, including non-generic cases. The main contributions are (i) a SageMath-based package that computes leading and higher-order diagonal asymptotics using Newton iteration for series expansions, (ii) explicit procedures for finding critical points on strata, minimal critical points, and contributing points, and (iii) practical improvements such as a Macaulay2 backend for Gröbner computations and a streamlined stratification workflow via transverse factorization. The results broaden the applicability of ACSV to non-smooth singular varieties and provide a robust tool for researchers to obtain precise asymptotics and constants for multivariate combinatorial sequences.
Abstract
The field of analytic combinatorics in several variables (ACSV) develops techniques to compute the asymptotic behaviour of multivariate sequences from analytic properties of their generating functions. When the generating function under consideration is rational, its set of singularities forms an algebraic variety -- called the singular variety -- and asymptotic behaviour depends heavily on the geometry of the singular variety. By combining a recent algorithm for the Whitney stratification of algebraic varieties with methods from ACSV, we present the first software that rigorously computes asymptotics of sequences whose generating functions have non-smooth singular varieties (under other assumptions on local geometry). Our work is built on the existing sage_acsv package for the SageMath computer algebra system, which previously gave asymptotics under a smoothness assumption. We also report on other improvements to the package, such as an efficient technique for determining higher order asymptotic expansions using Newton iteration, the ability to use more efficient backends for algebraic computations, and a method to compute so-called critical points for any multivariate rational function through Whitney stratification.