TROPICAL GEOMETRY Meaning and
Definition
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Tropical geometry is a branch of mathematics that studies algebraic varieties and their properties by using the framework of tropical semirings. The term "tropical" is derived from the tropical semiring, which is a set of numbers equipped with two arithmetic operations, namely tropical addition and tropical multiplication.
In tropical geometry, algebraic varieties are viewed as tropicalizations of classical algebraic varieties. The tropicalization process involves replacing the coefficients of the polynomials defining the variety with their tropical counterparts, which are computed using tropical arithmetic operations instead of the usual arithmetic operations. This transformation effectively erases the information about the exact values of the coefficients but retains important qualitative information about the variety's tropical structure.
Tropical geometry provides a geometric understanding of the behavior of algebraic varieties under tropicalization. It has proven to be a powerful tool for studying problems in various mathematical fields, including algebraic geometry, combinatorics, and optimization. The tropical approach often simplifies the analysis of certain algebraic phenomena, such as the study of linear systems, degenerations, and valuations, leading to new insights and solutions to longstanding problems.
Moreover, tropical geometry has connections with other mathematical disciplines, such as graph theory and matroid theory. Tropical graphs and tropical matroids are used to describe and analyze tropical varieties, providing a bridge between tropical geometry and these discrete mathematical structures.
Overall, tropical geometry offers a versatile and effective framework for understanding the tropical behavior of algebraic varieties and solving challenging mathematical problems.
Etymology of TROPICAL GEOMETRY
The word "tropical geometry" derives from the mathematical field called "tropical mathematics". The term "tropical" in this context comes from the use of max-plus algebra, which is a tropical semiring, where tropical operations such as addition and multiplication are defined as taking the maximum and minimum, respectively.
The origin of the term "tropical" can be traced back to Imre Simon, a Hungarian mathematician, who worked on optimization problems in the 1960s. Simon used the term "tropical" to describe certain types of optimization problems related to transportation networks in order to invoke a sense of exoticism and foreignness.
As tropical mathematics developed into a more widespread field, the term "tropical" became ubiquitous to describe various areas within it, including tropical geometry. Tropical geometry is a geometric approach to understanding tropical mathematics, which studies the behavior of tropical polynomials and tropical varieties.
