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The Gaussian entropy map in valued fields

Yassine El Maazouz

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.

The Gaussian entropy map in valued fields

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.

Paper Structure

This paper contains 12 sections, 12 theorems, 55 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction and notation
  2. Entropy of real multivariate Gaussian distributions
  3. Main results
  4. Background on valued fields and Gaussian measures
  5. Valued fields
  6. Local fields
  7. Lattices
  8. Gaussian measures
  9. The entropy map of local field Gaussian distributions
  10. Supermodularity of the entropy map
  11. The entropy map on non-archimedean fields
  12. Conclusion

Key Result

Theorem 1.3

The push-forward measure of a multivariate Gaussian measure on a local field by the valuation map is given by a tropical polynomial whose coefficients are given by the entropy map of this measure (see thm:trop_poly). Moreover, these coefficients are supermodular. The entropy map is still well define

Figures (4)

  • Figure 1: Tropical curve of $\varphi_\Lambda$ and its regular triangulation of the square for example \ref{['example:2d']}
  • Figure 2: Tropical geometry of the lattice $\Lambda$ for \ref{['example:3d']}.
  • Figure 3: The polyhedral complex $\mathop{\mathrm{trop}}\nolimits(\Lambda)$ for $\Lambda$ in \ref{['example:3d']}.
  • Figure 4: Intersections of $\mathcal{P}$ and $\mathcal{C}$ with the affine hyperplane $x+y+z+w + 1= 0$.

Theorems & Definitions (33)

  • Theorem 1.3
  • Remark 1.4
  • Example 2.2
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof
  • ...and 23 more