Universal types of singularities of solutions to functional equation systems

Abstract: Decompositions of combinatorial structures translate very often to functional equations with positive coefficients for their generating functions. A theorem by Bender says that if the generating function is univariate and the equation not linear, the generating function always has a dominant square root singularity - which in turn means that the the coefficients a(n) grow asymptotically at the rate c*n^(-3/2) R^n, where c and R are suitable constants. The result extends to strongly connected finite systems of equations, but as the system becomes infinite we can observe a broader variety of singularities appearing. In this talk, I will give an overview of functional equations systems and their singular behaviour in combinatorics and present some recent results on universal types of singularities of solutions to infinite systems which collapse to a single equation by introducing a second (catalytic) variable.

algebraic geometrynumber theory

Audience: researchers in the discipline

SFU NT-AG seminar

Series comments: The Number Theory and Algebraic Geometry (NT-AG) seminar is a research seminar dedicated to topics related to number theory and algebraic geometry hosted by the NT-AG group (Nils Bruin, Imin Chen, Stephen Choi, Katrina Honigs, Nathan Ilten, Marni Mishna).

We acknowledge the support of PIMS, NSERC, and SFU.

For Fall 2025, the organizers are Katrina Honigs and Peter McDonald.

We normally meet in-person in the indicated room. For online editions, we use Zoom and distribute the link through the mailing list. If you wish to be put on the mailing list, please subscribe to ntag-external using lists.sfu.ca