In this paper we prove, in a purely combinatorial-algebraic way, a structural quasi-polynomiality property for the Bousquet-Mélou–Schaeffer numbers. Conjecturally, this property should follow from the Chekhov–Eynard–Orantin topological recursion for these numbers (or, to be more precise, the Bouchard–Eynard version of the topological recursion for higher order critical points), which we derive in this paper from the recent result of Alexandrov–Chapuy–Eynard–Harnad. To this end, the missing ingredient is a generalization to the case of higher order critical points on the underlying spectral curve of the existing correspondence between the topological recursion and Givental’s theory for cohomological field theories.

在本文中，我们以纯组合代数的方式证明了Bousquet-Mélou-Schaeffer数的结构拟多项式性质。推测上，对于这些数字，此属性应遵循Chekhov-Eynard-Orantin拓扑递归（或更精确地说，对于高阶临界点，则是拓扑递归的Bouchard-Eynard版本），我们从本文得出Alexandrov–Chapuy–Eynard–Harnad的最新结果。为此，缺少的成分是对拓扑递归与同调场理论的纪梵特理论之间现有对应关系的基础光谱曲线上较高阶临界点的概括。