超几何多重和渐近性的一个证明
An ansatz for the asymptotics of hypergeometric multisums
摘要：
在组合数学、代数几何和微扰量子场论的计数问题中，经常出现由具有紧支集的超几何项的多和定义的序列。研究这类序列渐近展开式的标准方法是找到它们所满足的递推关系，将其转化为它们的生成级数所满足的微分方程，并分析复平面中的奇异性。我们提出了一个捷径，直接从结构的超几何长期的有限集，我们猜想（并在某些情况下证明），它包含所有的奇异性的生成系列。我们的建设，这个有限集是由一个平衡的系统的多项式方程的一个相当特殊的形式，让人想起贝特anastomy的解决方案。有限集也可以用势函数的临界值的集合来标识，以及用调节器函数对加性$K$ -理论群的元素的求值来标识。我们在一些特殊的情况下给出了我们的猜想的证明，我们用大量的例子来说明我们的结果。
英文摘要：
Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singulatiries in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive $K$-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.
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