论文标题
较高时刻的本地与全球原则上的数字字段
Local to global principle over number fields for higher moments
论文作者
论文摘要
密度的本地与全球原理是Poonen和Stoll提出的一种非常方便的工具，以计算给定整数子集的密度。在本文中，我们提供了一个有效的标准，可以找到一个有限尺寸的自由模块的子集的密度（例如，平均值，差异）的所有较高矩（均值，差异）。更确切地说，我们提供了一个本地到全球原则，该原理允许在一般数字字段$ k $上计算与密度相对应的所有较高矩。这项工作通过两种方式来促进对密度计算的本地原理的理解：一方面，它扩展了明亮，褐变和拉夫兰的结果，在那里，它们为数量字段的密度提供了本地至全球原理；另一方面，它将最近的结果扩展到全球原理，即整数的预期值，以延伸到代数整数环，并将矩高到高于期望值。为了显示我们的方法的有效性和适用，我们计算了爱森斯坦多项式的密度，平均值和方差，并在数量字段上转移了爱森斯坦多项式。这扩展了（并完全覆盖）通过临时方法获得的文献。
The local to global principle for densities is a very convenient tool proposed by Poonen and Stoll to compute the density of a given subset of the integers. In this paper we provide an effective criterion to find all higher moments of the density (e.g. the mean, the variance) of a subset of a finite dimensional free module over the ring of algebraic integers of a number field. More precisely, we provide a local to global principle that allows the computation of all higher moments corresponding to the density, over a general number field $K$. This work advances the understanding of local to global principles for density computations in two ways: on one hand, it extends a result of Bright, Browning and Loughran, where they provide the local to global principle for densities over number fields; on the other hand, it extends the recent result on a local to global principle for expected values over the integers to both the ring of algebraic integers and to moments higher than the expected value. To show how effective and applicable our method is, we compute the density, mean and variance of Eisenstein polynomials and shifted Eisenstein polynomials over number fields. This extends (and fully covers) results in the literature that were obtained with ad-hoc methods.
