A case study in bivariate singularity analysis

The multivariate singularity analysis of Pemantle and Wilson is explored and then used to derive an asymptotic expression for the number of bicolored supertrees, realized as the diagonal sequence of a bivariate rational function F. These asymptotics have been obtained previously by univariate methods, but the analysis contained herein serves as a case study for the general multivariate method. The analysis itself relies heavily on the structure of a height function h along the pole set V of F. What makes this example interesting is the geometry of h on V, namely that h has a degenerate saddle point away from the boundary of the domain of analyticity of F that contributes to the asymptotic analysis. Due to the geometry of h at this point, the coefficient analysis can not be computed directly from the standard formulas of multivariate singularity analysis. Performing the analysis in this case represents a first step towards understanding general cases of this geometric type.