Effective Asymptotics of Combinatorial Systems

Analytic combinatorics studies asymptotic properties of families of combinatorial objects using complex analysis on their generating functions. In their reference book on the subject, Flajolet and Sedgewick describe a general approach that allows one to derive precise asymptotic expansions starting from systems of combinatorial equations. In the situation where the combinatorial system involves only cartesian products and disjoint unions, the generating functions satisfy polynomial systems with positivity constraints for which many results and algorithms are known. We extend these results to the general situation. This produces an almost complete algorithmic chain going from combinatorial systems to asymptotic expansions. Thus, it is possible to compute asymptotic expansions of all generating functions produced by the symbolic method of Flajolet and Sedgewick when they have algebraic-logarithmic singularities (which can be decided), under the assumption that Schanuel's conjecture from number theory holds. That conjecture is not needed for systems that do not involve the constructions of sets and cycles.