  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.
