Three lectures on tropical algebra

TL;DR

This paper provides an overview of tropical algebra, exploring tropical polynomials, ideals, and their geometric connections, along with algebraic constructions like matrix and Clifford algebras in the tropical setting.

Contribution

It introduces new perspectives on tropicalization, including a refined limit theorem and the tropicalization of algebraic structures such as exterior, matrix, and Clifford algebras.

Findings

Refined understanding of Berkovich analytification and tropicalization via bend relations.

Tropicalization of algebraic structures like exterior, matrix, and Clifford algebras.

New insights into tropical Plücker relations and Morita theory in tropical algebra.

Abstract

This document is a slightly expanded version of a series of talks given by J. Giansiracusa at the workshop `Geometry over semirings' at Universitat Aut\`{o}noma de Barcelona in July 2025. In the first lecture we introduce tropical polynomials, ideals, congruences, and how the connection with tropical geometry is made via congruences of bend relations. Tropical geometry and matroid theory are telling us that we should focus attention on a narrow slice of the world of tropical algebra, and this leads to the theory of tropical ideals (as developed by Maclagan and Rinc\'{o}n) and an abundance of interesting open questions. In the second lecture we examine the relationship between Berkovich analytification and tropicalization from the perspective of bend relations, giving a refinement of Payne's influential limit theorem. In the third lecture we set aside geometry and focus on tropicalization via bend relations as a construction in commutative and non-commutative algebra. Constructions such as symmetric algebras, exterior algebras, matrix algebras, and Clifford algebras can be tropicalized. In the case of exterior algebras, the resulting tropical notion beautifully completes the picture of the Pl\"{u}cker embedding and gives a new perspective on the tropical Pl\"{u}cker relations. For matrix algebras and Clifford algebras, Morita theory becomes an interesting topic.