For me, the topic of tropicalization is new and I am trying to understand what new insights and perspectives tropicalization could provide me on the following two curves (degree 6 and 8). So it is a pure experiment without knowing what can come out of it. Both curves construct a "heartshape" (see this figure), about which I would like to learn experimentally what information tropicalization can contribute to me.
With the kind help of Jan-Magnus Økland (see this MSE post) I could computationally tropicalize two curves:
- $\color{red}{\textbf{Red:}}$ $0=16y^6+48x^2y^4-20y^4+48x^4y^2-40x^2y^2+5y^2+16x^6-20x^4+5x^2-x$ (curve of degree)
- $\color{blue}{\textbf{Blue:}}$ $0=-64y^8-256x^2y^6+112y^6-384x^4y^4+336x^2y^4-56y^4-256x^6y^2+336x^4y^2-112x^2y^2+7y^2-64x^8+112x^6-56x^4+7x^2+x$ (curve of degree 8)
Now I am trying to interpret the result of this tropicalization, e.g. how do I "read" these tropical curves? For this I watched the video "What is tropical geometry?", but it is still for me not very clear how to comprehend that principle in the context of my concrete case. I would be grateful for any directions and hints to interpret the tropicalization result.
For those who are interested - I used the following Singular code for the tropicalization:
LIB "tropical.lib";
ring r=(0,t),(x,y),dp;
poly f=16*y^6+48*x^2*y^4-20*y^4+48*x^4*y^2-40*x^2*y^2+5*y^2+16*x^6-20*x^4+5*x^2-x;
poly f=-64*y^8-256*x^2*y^6+112*y^6-384*x^4*y^4+336*x^2*y^4-56*y^4-256*x^6*y^2+336*x^4*y^2-112*x^2*y^2+7*y^2-64*x^8+112*x^6-56*x^4+7*x^2+x;
drawTropicalCurve(f,"max");