Three lectures on tropical algebra

TL;DR

This work surveys how bend relations unify tropical algebra with tropical geometry, Berkovich analytification, and noncommutative algebra. It develops a scheme-theoretic view of tropicalization, links valuations to tropical linear spaces via valuated matroids, and demonstrates how tropical ideals provide a robust framework beyond realizable tropicalizations. The exposition extends tropicalization to noncommutative settings (matrix and Clifford algebras) and revisits classical constructs like Plücker embeddings through a tropical lens, including a tropical Plücker theory and Morita-type questions. Overall, the paper highlights foundational structures (tropical ideals, Dressians, and tropical linear spaces) and outlines key open questions about realizability, gluing of tropical schemes, and noncommutative tropical geometry with potential connections to Bott periodicity and Morita theory.

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 Autònoma 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 Rincón) 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 Plücker embedding and gives a new perspective on the tropical Plücker relations. For matrix algebras and Clifford algebras, Morita theory becomes an interesting topic.

Figures (5)

• Figure 1: Family of semirings converging to $\mathbb{T}$.
• Figure 2: The graph of the tropical Laurent polynomial $f(X)= (X^2)\oplus (2)\oplus (8\odot X^{-2})$ is shown in red. It is the minimum of the linear functions represented by its three monomial terms.
• Figure 3: The graphs of $f=1 \oplus x$ and $f^{-1}$.
• Figure 4: Ilustration of $V^\textup{trop}(f)$.
• Figure 5: The Berkovich projective line $(\mathbb{P}^{1})^{\mathop{\mathrm{an}}\nolimits}$, adapted from an illustration of Joe Silverman.

Theorems & Definitions (43)

• Example 1.2.1
• Example 1.2.2: Tropical semiring
• Example 1.2.3
• Example 1.2.4: Models for the tropical semiring
• Example 1.2.5: $\mathbb{T}$ as a limit of semirings, Shaw_2015
• Example 1.3.1
• Example 1.5.1
• Remark 1.5.2
• Definition 1.6.1
• Example 1.6.2