We’re finally at the point where we can provide the first definition of Tropical Geometry, and for the sake of personal historicity, it will be the one that I didn’t particularly like when I learned of it.
Remember that the point of the algebraic geometry, as it’s studied, is that we study a geometric object by studying the functions defined on that object. The two perspectives are equivalent, and you can use tools from one to study the other (and of course, vice versa).
As an aside, this is one way that you can understand what is meant by non-commutative geometry, at least coarsely. One fact about all of the geometric objects that we study is that the rings of functions defined on any of these things are what we call commutative. That is, for every pair of functions . If you think about this, this is really a reflection of the fact that at it’s heart, the functions take values in real or complex numbers (well…), and so since those satisfy , it follows that functions into them do as well.
So where does non-commutative geometry come in? Well, what happens when you study non-commutative rings? What do they represent “geometrically”?
The point in this case is that if we expand generalize the left-hand side of the equivalence
commutative -algebras geometric stuff
then this in some sense should provide something that is a generalization of the right-hand side as well. This is a well trod-upon tactic in mathematics, and provides us with notions such as a stack (note: not the same as a stack in the computer science world!), or derived schemes, or even derived stacks (combine the two).
Anyhow! So I promised that I would talk about Tropical geometry, and how it fits into the picture. Well, here goes.
See, a ring is something that satisfied a collection of proprties (or axioms) which state how we can multiply and add things together. These basically mean that they behave like the familiar integers, real numbers, or whatever—they look like the normal number systems that we’re all familiar with. It turns out that this list of properties is all we really need to build a phenomenally rich geometric world.
So for Tropical geometry, we look at a slightly different starting point. Consider the real numbers, but with the following funny rules for “addition” and “multiplication”:
Ok, what the hell is this. Multiplication is addition now? Addition is… the maximum? This seems very strange (and it is!) but it turns out that with this bizarre notion of addition and multiplication that we still get a surprising amount of similar properties than normal addition and multiplication have. For example, we still have that
i.e. the distributive law. We also have a multiplicative identity ( for every ), we have multiplicative inverses (since , the identity). We even can have an additive identity if we include in the package. What we can’t have though, is additive inverses and hence no subtraction.
So yeah, weird. Something which satisfies this collection of rules is a semi-ring, and with this in mind, we do exactly what you should be now expecting: Tropical geometry is geometry done using semi-rings.