Contemporary Mathematics Volume 580, 2012 Introduction to tropical algebraic geometry Diane Maclagan Abstract. This is an expository introduction to tropical algebraic geometry based on my lectures at the Workshop on Tropical Geometry and Integrable Systems in Glasgow, July 4–8, 2011, and at the ELGA 2011 school on Algebraic Geometry and Applications in Buenos Aires, August 1–5, 2011. 1. Introduction Tropical algebraic geometry is algebraic geometry over the tropical semiring (Definition 1.1). This replaces an algebraic variety by a piecewise linear object which can be studied using polyhedral combinatorics. Tropical geometry has exploded as an area of research in the last decade, with many new connections and applications appearing each year. These include enu- merative geometry, mirror symmetry, arithmetic geometry, and integrable systems. It builds on the older area of tropical mathematics, more commonly known as max- plus algebra, which arises in semigroup theory, computer science, and optimization. The name “tropical” was coined by some French mathematicians in honor of the Brazilian computer scientist Imre Simon. See [6] or [19] for an introduction to this older area. The goal of this expository and elementary article is to introduce this exciting new area. We develop the theory of tropical varieties and outline their structure and connection with “classical” varieties. There are several approaches to tropical geometry. We follow the “embedded” approach, which focuses on tropicalizing classical varieties. Another important branch of the subject focuses on developing an abstract theory of tropical varieties in their own right. See work of Mikhalkin and collaborators [29], [30] for details on this. This direction is most developed for curves. One of the attractions of tropical geometry is that it has so many disparate, but connected, facets, so any survey is necessarily incomplete. We begin by introducing tropical mathematics, focusing on tropical polynomials and their solutions. In this paper we will follow the minimum convention for the tropical semiring: Definition 1.1. The tropical semiring is R {∞}, with operation and given by a b = min(a, b) and a b = a + b. 2010 Mathematics Subject Classification. Primary 14T05 Secondary 14M25, 52B20, 12J25. Partially supported by EPSRC grant EP/I008071/1. c 2012 American Mathematical Society 1
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