# Tropical varieties

This page is still under heavy construction. Expect massive updates.

# Tropical polynomials and regular tropical functions

Each tropical polynomial ^{[1]} For example, the previous tropical polynomial and the tropical polynomial

In other words, each define the function given piecewise by

In any case, observe the function defined by a tropical polynomial is always a piecewise linear function. This is one instance of why tropical geometry is sometimes referred to as "piecewise linear" or "semi-linear" geometry.

# Bend loci

In classical algebraic geometry, the geometric objects are the zero sets of polynomials. For instance, the plane parabola defined by

In tropical geometry, however, we don't usually consider zero sets of tropical polynomials. This is for a variety of reasons. First, by "zero" it's not clear if we mean "the real number 0" or "the tropical number **bend loci** (or **double-max loci**) of tropical polynomials. These are the regions where the graph of the tropical polynomial

It might still seem strange why these bend loci are the focus of much of tropical geometry. The honest reason is the tropical algebraic geometry first arose through an idea of **tropicalization**, where classical algebraic varieties were "tropicalized" via a family of logarithmic maps. When this idea was first investigated, it was discovered that the images of classical algebraic varieties (at least curves in the complex plane) were exactly regions in the tropical plane (originally, simply the real plane) matching bend loci of tropical polynomials. Because of this, it seemed reasonable to consider these bend loci as the images of classical varieties, and hence worthy of study in their own right.

# Congruence varieties

There is another type of geometric object one can reasonably study. Returning for the moment to the example of the classical plane parabola defined by

In the tropical setting, however, this is no longer true. While one can certainly define the locus of points *these* types of geometric subsets behave. Should they be called tropical varieties? Or should the bend loci? Personally, I think neither should be. I think we should stick to the terms **bend varieties** and **congruence varieties**. We'll see if that ever catches on, though.

In view of this, it's probably best to think of evaluation as defining a map from the semiring of tropical polynomials to the semiring of "regular" tropical functions. β©οΈ