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.

多元有理函数的系数序列出现在组合数学的许多领域。它们的对角系数序列具有良好的算术和渐近性质，并且在多个变量（ACSV）中的解析组合域使计算渐进展开成为可能。我们从有效性的角度考虑这些方法。特别是，在给定有理函数的情况下，ACSV要求确定一个（通常）有限的点集合，这些集合称为关键点和最小点。临界度是一个代数条件，这意味着它可以通过计算机代数中的经典方法很好地处理，而极小度是一个半代数条件，它描述了多元幂级数的收敛域边界上的点。我们展示了如何在某些通用性假设下使用符号数字技术获得多元有理函数对角线系数序列的显性渐近性。就我们所知，这是第一个对任意数量的有理函数进行渐近枚举的全自动处理和复杂度分析。