The coefficient sequences of multivariate rational functions appear in many areas of combinatorics. Their diagonal coefficient sequences enjoy nice arithmetic and asymptotic properties, and the field of analytic combinatorics in several variables (ACSV) makes it possible to compute asymptotic expansions. We consider these methods from the point of view of effectivity. In particular, given a rational function, ACSV requires one to determine a (generically) finite collection of points that are called critical and minimal. Criticality is an algebraic condition, meaning it is well treated by classical methods in computer algebra, while minimality is a semi-algebraic condition describing points on the boundary of the domain of convergence of a multivariate power series. We show how to obtain dominant asymptotics for the diagonal coefficient sequence of multivariate rational functions under some genericity assumptions using symbolic-numeric techniques. To our knowledge, this is the first completely automatic treatment and complexity analysis for the asymptotic enumeration of rational functions in an arbitrary number of variables.

中文翻译：

---

通过多项式系统的半数值算法的多元有理函数的有效系数渐近

多元有理函数的系数序列出现在组合学的许多领域。它们的对角系数序列具有很好的算术和渐近特性，并且多变量分析组合 (ACSV) 领域使得计算渐近展开成为可能。我们从有效性的角度考虑这些方法。特别是，给定一个有理函数，ACSV 需要确定一个（一般）有限的点集合，这些点称为临界点和最小点。临界性是一种代数条件，这意味着它可以被计算机代数中的经典方法很好地处理，而极小性是描述多元幂级数收敛域边界上的点的半代数条件。我们展示了如何使用符号数字技术在一些通用假设下获得多元有理函数的对角系数序列的主导渐近线。据我们所知，这是对任意数量变量中的有理函数的渐近枚举的第一个完全自动处理和复杂性分析。