SFU Number Theory and Algebraic Geometry Seminar: Eva-Marie Hainzl

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.