The almost-entropic regions are not semialgebraic

TL;DR

This paper proves that the almost-entropic region of order four cannot be described by polynomial inequalities, showing it is not semialgebraic, and confirms it is not polyhedral, extending Matus's theorem.

Contribution

It demonstrates that the almost-entropic region of order four is not semialgebraic, providing new insights into its geometric complexity.

Findings

The almost-entropic region of order four is not semialgebraic.

This region is not polyhedral.

The result extends Matus's theorem.

Abstract

We prove that the almost-entropic region of order four is not semialgebraic, we get as a corollary the well-known Theorem of Matus, which asserts that this region is not polyhedral