This following is a brief summary of our research meeting on 2024-07-19.

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 $f(x)=x^{\odot 2}\oplus x\oplus 1$ induces a $\mathbf{T}$-valued function on the tropical line, which in real operations is $f(x)=max\{2x,x,1\}$.
• Unlike polynomials in $\mathbf{R}[x]$, 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 $x$.
• 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 $x$ 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 double-max loci) for a tropical polynomial in two variables $x$ and $y$.

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 $f(x)=g(x)$ for some tropical polynomials $f(x)$ and $g(x)$. 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 $f(x,y)=g(x,y)$ for some tropical polynomials $f(x,y)$ and $g(x,y)$. Try to find examples where:

  • The polynomials are both of low degree, say less than 2.
  • The polynomials are both of degree 2.