Local to global principle over number fields for higher moments

TL;DR

This paper develops a comprehensive local to global principle for calculating all higher moments of densities of subsets in modules over algebraic integers of number fields, extending previous results and demonstrating practical applications.

Contribution

It introduces an effective criterion for computing all higher moments of density over number fields, extending prior local to global principles to include higher moments beyond the expected value.

Findings

Computed density, mean, and variance of Eisenstein polynomials over number fields.

Extended local to global principles to higher moments and algebraic integer rings.

Provided a unified, effective method surpassing ad-hoc approaches in the literature.

Abstract

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.