标题： 组合系统的有效渐近分析 显示英文标题

标题： Effective Asymptotics of Combinatorial Systems

摘要： 分析组合学使用生成函数上的复分析来研究组合对象族的渐近性质。 在他们关于该主题的参考书中，Flajolet 和 Sedgewick 描述了一种通用方法，该方法允许从组合方程组中推导出精确的渐近展开式。 在组合系统仅涉及笛卡尔积和不相交并的情况下，生成函数满足具有正性约束的多项式系统，对此已有许多结果和算法。 我们将这些结果扩展到一般情况。 这产生了一个几乎完整的算法链，从组合系统到渐近展开式。 因此，当 Flajolet 和 Sedgewick 的符号方法产生的生成函数具有代数-对数奇点时（这是可以判断的），可以计算所有生成函数的渐近展开式，前提是数论中的 Schanuel 猜想成立。 对于不涉及集合和环构造的系统，不需要这个猜想。 显示英文摘要

摘要： 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.