Tropical Geometry, part 2.

So last post we went over the origin of the name “Tropical Geometry”, but not what it was. I would like to start to do that, but I think that in order to do so we have to take a few steps back and understand a little bit about algebraic geometry as a whole.

The idea of algebraic geometry is to study the geometry of objects defined by algebra. Let’s look at the simplest non-trivial example. As you may recall from high school mathematics, a circle of radius R in the plane can be seen as the set of all solutions to the equation

x^2 + y^2 - R^2 = 0

although I have perhaps written it somewhat idiosyncratically, with all of the terms on the left-hand side of the equals sign. The point is that a circle can be defined by a polynomial equation, and these are the objects that interest us: those geometric figures that can be described by polynomial equations (this is the algebra part of algebraic geometry).

By contrast, if we consider the graph of the function y = e^x, then there is no algebraic equation that the coordinates of this graph will satisfy, and so it is not an object that we are interested in in this context.

So how does one study these? Well, it turns out that a major insight was that you can study objects (geometric or otherwise) by studying all of the functions that are defined on those objects. In our case since we are concerned with—for the time being—figures that are cut out by polynomials in the plane, we are also going to restrict ourselves to considering polynomial-type functions defined on these objects. So what are those?

Well, an obvious source of such a function is any polynomial in the variables x, y. Since our circle lies in the plane, any function defined on the plane a fortiori will define a function on our circle: just define the value of the function on the circle to be the value of the planar function at that point.

The problem with this approach is that you will typically get too many functions. There may be more than one function defined on the plane whose values on the circle are the same! For example, the two polynomials

f(x, y) = x^2


g(x,y) = -y^2 + R^2

will secretly yield the exact same function on our circle. The reason is that f(x, y) - g(x, y) = x^2 + y^2 - R^2—but this is the defining equation of our circle! So what we should do is say that any two functions on the plane are, for our purposes, the same function if they differ by the defining equation of our geometric figure. It turns out that if we do this, then we can get a meaningful way to talk about all of the functions on our figure.

Moreover—and this shouldn’t necessarily be obvious—in a certain sense, one can show that if we do this, then the geometric figure is entirely equivalent to the so-obtained functions. That is, it is completely equivalent to study either the figure itself, or the functions as we have described them. This is a very powerful shift in perspective.


About charlesflorian

Mathematician. Climber. Living in Copenhagen.
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