Entropy algebras and Birkhoff factorization

TL;DR

This paper introduces Rota-Baxter structures and Birkhoff factorizations within min-plus semirings and their thermodynamic deformations, connecting algebraic, quantum information, and number theory concepts.

Contribution

It extends Rota-Baxter and Birkhoff factorization frameworks to min-plus semirings and their deformations, including quantum entropy measures, with diverse applications.

Findings

Developed Rota-Baxter structures for min-plus semirings

Established Birkhoff factorizations in thermodynamic deformations

Applied framework to algebraic and quantum information contexts

Abstract

We develop notions of Rota-Baxter structures and associated Birkhoff factorizations, in the context of min-plus semirings and their thermodynamic deformations, including deformations arising from quantum information measures such as the von Neumann entropy. We consider examples related to Manin's renormalization and computation program, to Markov random fields and to counting functions and zeta functions of algebraic varieties.