Introduction to tropical algebraic geometry

✨ Summary

The paper gives an elementary introduction to tropical algebraic geometry, where ordinary addition and multiplication are replaced by the minimum and addition operations of the tropical semiring. Tropical polynomials become piecewise-linear functions, and their solution sets are nondifferentiability loci or balanced polyhedral complexes.

It defines valuations and tropicalizations of Laurent polynomials and algebraic varieties, explaining how tropical varieties encode valuation data from classical varieties. The Fundamental Theorem identifies a tropical variety with the Euclidean closure of the coordinatewise valuations of points on the original variety. The Structure Theorem shows that tropicalizations of irreducible varieties are pure-dimensional, weighted, balanced, rational polyhedral complexes connected through codimension one.

The paper also explains how plane tropical curves are constructed from Newton polytopes and regular subdivisions, and how tropical varieties can be computed using initial ideals and Gröbner complexes. It concludes by relating tropical varieties to toric varieties and tropical compactifications, including the moduli spaces of stable marked curves and the tropical moduli space of phylogenetic trees.

The paper is an expository contribution rather than a new algorithmic or theoretical result, but it helped make the fundamental definitions and theorems of tropical geometry accessible to researchers entering the field. Its themes continued into later developments such as tropical ideals, which provide an algebraic framework for tropical subschemes and extend the treatment of tropicalizations beyond polyhedral sets. (arxiv.org) The broader subject has since been consolidated in the authors’ 2015 textbook and applied across algebraic geometry, combinatorics, moduli theory, optimization, phylogenetics, and related areas; no specific industrial deployment directly attributable to this paper was identified. (old.maa.org)