REU Meeting  20240719
This following is a brief summary of our research meeting on 20240719.
Summary of discoveries
Tropical polynomials and functions on tropical space
 We talked about how tropical polynomials induce functions on tropical space (by usual evaluation). For example, the tropical polynomial
induces a valued function on the tropical line, which in real operations is .  Unlike polynomials in
, however, tropical polynomials are not uniquely identified by the functions they induce. To put it another way, two different tropical polynomials can define the exact same function on tropical space.  For a given tropical polynomial, there is an entire collection (really an equivalence class!) of tropical polynomials that induce the same function on tropical space.
Graphs of tropical functions
 We took a look at some examples of sketching the graphs of tropical polynomials in one variable
.  We saw that the graph is always made of a union of line segments (and rays) with slopes that can only increase as we move in the positive
direction.  We saw how the key features of the graph are: 1) where it "bends"; and 2) the slope of each piece.
 We recalled how the tropicalizations of complex curves led shapes that were exactly bend loci of tropical polynomials.
 We outlined a strategy for sketching bend loci (aka doublemax loci) for a tropical polynomial in two variables
and .
Tropical tangency
 We used Desmos to interactively investigate the idea of tropical tangency a bit.
 We noted that the locus of points dual to tropical tangent lines might be something a bit different from a bend locus ...
Tasks for next meeting
We will now investigate a new type of tropical subset, namely congruence varieties. These are subsets in tropical space where two tropical polynomials agree.

First create a few examples in one variable, finding all points in the tropical line where two different tropical polynomials agree; i.e., satisfy an equality
for some tropical polynomials and . Try to find examples where:  The congruence variety consists only of discrete points
 The congruence variety contains at least one line segment or ray

Now try the same thing in two variables, finding all points in the tropical plane where two different tropical polynomials agree; i.e., satisfy an equality
for some tropical polynomials and . Try to find examples where:  The polynomials are both of low degree, say less than 2.
 The polynomials are both of degree 2.
Every bend locus is the union of a bunch of congruence varieties. Can you show this? (Try a few examples!)