Characteristic one, entropy and the absolute point

TL;DR

This paper explores the mathematical structure of characteristic one, connecting it to tropical geometry, entropy, and the absolute point, and extends classical constructions to this new setting with applications to zeta functions.

Contribution

It introduces the notion of perfect semi-rings of characteristic one and adapts Witt ring constructions, linking them to entropy and providing new insights into F_1-schemes and zeta functions.

Findings

Connection between characteristic one and entropy/thermodynamics

Extension of Witt ring construction to characteristic one

Computation of zeta functions for F_1-schemes and elliptic curves

Abstract

We show that the mathematical meaning of working in characteristic one is directly connected to the fields of idempotent analysis and tropical algebraic geometry and we relate this idea to the notion of the absolute point. After introducing the notion of "perfect" semi-ring of characteristic one, we explain how to adapt the construction of the Witt ring in positive characteristic to the limit case of characteristic one. This construction unveils an interesting connection with entropy and thermodynamics, while shedding a new light on the classical Witt construction itself. We simplify our earlier construction of the geometric realization of an F_1-scheme and extend our earlier computations of the zeta function to cover the case of F_1-schemes with torsion. Then, we show that the study of the additive structures on monoids provides a natural map from monoids to sets which comes close to fulfill the requirements for the hypothetical curve compactifying Spec Z over the absolute point. Finally, we test the computation of the zeta function on elliptic curves over the rational numbers.