A SageMath Package for Analytic Combinatorics in Several Variables: Beyond the Smooth Case

2025-04-14

AI Summary

This work addresses the failure of asymptotic analysis for multivariate generating functions whose singular varieties are nonsmooth—a longstanding limitation in analytic combinatorics. We introduce the first rigorous asymptotic framework capable of handling nonsmooth algebraic singularities. Methodologically, we systematically integrate Whitney stratification theory with Analytic Combinatorics in Several Variables (ACSV), leveraging SageMath-based symbolic computation, Newton iteration, and algorithms from computational algebraic geometry to enable automatic critical point detection, high-order asymptotic expansion, and efficient algebraic backend support. Our key contribution is the extension of ACSV beyond its traditional restriction to smooth singularities: within the `sage_acsv` package, we implement a substantive generalization that supports fully automated derivation of leading and higher-order asymptotic terms for nonsmooth cases. This advancement significantly enhances the applicability, robustness, and automation level of asymptotic analysis for multivariate sequences.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Cognitive Modeling & Cognitive Systems: Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Computational social science

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.

Problem

• Develops software for multivariate sequence asymptotics with non-smooth singularities
• Extends ACSV methods using Whitney stratification of algebraic varieties
• Enhances sage_acsv package with higher-order expansions and critical point computation

Innovation

• Whitney stratification for non-smooth singular varieties
• Newton iteration for higher order asymptotics
• Critical points computation via Whitney stratification