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Updated: Jan 7, 2026
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Magnetically Induced Rotating Rayleigh-Taylor Instability
Published on: March 3, 2017
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Universality for Taylor coefficients of rational functions under perturbations
Yuliy Baryshnikov1,2, Robin Pemantle3
1Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801.
Summary
We developed computational methods for analytic combinatorics in several variables. These methods determine the asymptotic behavior of coefficients for rational generating functions, simplifying analysis of statistical physics models.
Area of Science:
- Computational mathematics
- Analytic combinatorics in several variables
- Algebraic combinatorics
Background:
- Analytic combinatorics is crucial for analyzing combinatorial structures.
- Rational generating functions are widely used in discrete mathematics and physics.
- Cluster algebras provide complex recursive structures.
Purpose of the Study:
- Introduce novel computational methods for analytic combinatorics in several variables.
- Analyze rational generating functions with specific dominant singularity conditions.
- Derive asymptotic formulas for coefficients of these functions.
Main Methods:
- Developing computational techniques for analytic combinatorics.
- Analyzing rational generating functions near their dominant singularities.
- Applying methods to recursions from cluster algebras and statistical physics models.
Main Results:
- Established conditions for dominant singularities of rational generating functions.
- Showed asymptotic coefficients are determined by the leading homogeneous term of the denominator.
- Demonstrated asymptotic behavior for statistical physics models is described by elliptic and hyperelliptic integrals.
Conclusions:
- The developed methods provide efficient computation for complex combinatorial problems.
- The findings simplify the asymptotic analysis of coefficients in rational generating functions.
- Elliptic and hyperelliptic integrals offer a pathway for computing asymptotic behaviors in statistical physics.
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