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.

翻译：解析组合学通过对其生成函数进行复分析，研究组合对象族的渐近性质。在该领域的权威著作中，Flajolet与Sedgewick提出了一种通用方法，可从组合方程组出发推导精确的渐近展开式。当组合系统仅涉及笛卡尔积与无交并运算时，其生成函数满足具有正性约束的多项式方程组——针对此类情形已有诸多结果与算法。本文将上述结果推广至一般情形，由此构建了从组合系统到渐近展开的近乎完整的算法链条。因此，在假定数论中的沙努尔猜想成立的前提下（该猜想对于不涉及集合与循环构造的系统非必需），当生成函数具有代数-对数型奇点（此性质可判定）时，能够计算所有由Flajolet与Sedgewick符号方法生成的函数的渐近展开。