Toroidal methods for computing zeta functions of groups and rings

最初在20世纪80年代引入子群增长领域，对群和环的zeta函数的研究已经发展成为一个深入的理论，结合了代数，组合数学，代数几何和其他数学领域的方法。然而，这种zeta函数的显式计算仍然极具挑战性。这个项目的主要目标是发展计算和分析群和环的zeta函数的环形方法。更确切地说，我们考虑的zeta函数允许欧拉乘积因子分解到由素数索引的局部zeta函数中，我们试图计算这些局部因子。我们的第一个主要目标是开发和实现一个算法，用于计算非退化假设下的局部zeta函数相对于某些相关的牛顿多面体。我们的算法将联合收割机代数几何计算和凸几何的方法相结合。我们的第二个主要目标是发展的方法，研究分析性质的本地zeta功能的群体和环，再次根据非退化的假设。特别是，我们将开发的方法，用于研究当地的极点频谱等zeta函数。我们的项目的第三个主要目标是应用我们的方法来研究有趣和具有挑战性的群和环类的zeta函数。这些计算都将刺激我们的算法的进一步发展，他们将提供一个试验场在该地区的成果。作为该项目一部分开发的所有软件和数据库将免费提供。

Originally introduced in the 1980s in the area of subgroup growth, the study of zeta functions of groups and rings has since evolved into a deep theory that combines methods from algebra, combinatorics, algebraic geometry, and other areas of mathematics. The explicit computation of such zeta functions, however, remains extremely challenging. The main objective of this project is to develop toroidal methods for computing and analyzing zeta functions of groups and rings. More precisely, the zeta functions that we consider admit Euler product factorisations into local zeta functions indexed by primes and we seek to compute these local factors. Our first main goal is to develop and implement an algorithm for computing such local zeta functions under non-degeneracy assumptions with respect to certain associated Newton polyhedra. Our algorithm will combine algebro-geometric computations and methods from convex geometry. Our second main goal is to develop methods for studying analytic properties of local zeta functions of groups and rings, again under non-degeneracy assumptions. In particular, we will develop methods for studying the local pole spectra of such zeta functions. The third main goal of our project is to apply our methods to study zeta functions of interesting and challenging classes of groups and rings. These computations will both stimulate further developments of our algorithms and they will provide a testing ground for conjectures in the area. All software and databases developed as part of this project will be made freely available.