The Gaussian entropy map in valued fields

TL;DR

This paper extends the concept of the entropy map for multivariate Gaussian distributions to local and non-archimedean valued fields, characterizing its image and properties.

Contribution

It introduces the Gaussian entropy map for valued fields, proves its supermodularity, and explicitly computes its image in three dimensions.

Findings

The entropy map's image lies in the supermodular cone.

The map determines the distribution of the valuation vector.

Explicit computation of the image in dimension three.

Abstract

We exhibit the analog of the entropy map for multivariate Gaussian distributions on local fields. As in the real case, the image of this map lies in the supermodular cone and it determines the distribution of the valuation vector. In general, this map can be defined for non-archimedian valued fields whose valuation group is an additive subgroup of the real line, and it remains supermodular. We also explicitly compute the image of this map in dimension 3.