# Some Basic Properties of Tropical Amoebas

I am working through Sturmfels' new book on Tropical Geometry with another student, and we are stuck at a pretty important concept, namely the basic properties of amoebas. Let me reproduce a bit of the material from this section (page 21).

So we take an ideal $I$ in the ring of Laurent polynomials over $\mathbb{C}$, i.e. a polynomial in $S = \mathbb{C}[x_1^{\pm 1}, x_2^{\pm 1}, \dots, x_n^{\pm 1}]$ and consider its variety in $(\mathbb{C}^*)^n$. Then we define the amoeba of $I$ to be the set $$\mathcal{A}(I) = \lbrace (log(|z_1|), log(|z_2|), \dots, log(|z_n|)) \in \mathbb{R}^n : \textbf{z} = (z_1, z_2, \dots, z_n) \in V(I) \rbrace$$

i.e. the component-wise logarithm. I understand that it's unbounded, has analytic (properly) embedded manifolds as boundary components and that the components of the complement will be convex in the max-convention, concave in the min-convention. In the book, no properties are proven to hold for these objects and one is simply directed to exceptionally technical papers. Let's switch only to curves to simplify matters.

The three basic properties which we don't understand are the following:

(1) Why is it true that the amoeba has only finitely many tentacles? And, (2) why does the thickness of a given tentacle go to zero? I understand that the components of the complement are strictly convex, but it's not enough of course. And, probably the most technical, (3) why does $\mathcal{A}(I)$ cover the tropicalization of the curve? The "tropicalization" of a curve takes a curve $$\sum \alpha_{i,j}x^iy^j \xrightarrow{tropic} \bigoplus x^i \odot y^j$$

where we keep the terms with $\alpha_{i,j} \neq 0$.

Presumably at least some of these are difficult. I am more interested in a heuristic at this point in time. Does anybody have an explanation for these phenomena? Thanks in advance!

And of course, if somebody knows a reference that treats this material for a non-specialist, that would be even better.

• There is something called "tropical amoebas". You learn something new every day..."tentacles"?? – Olórin Apr 2 '16 at 0:48
• Yeah, they're pretty cool! I hope! Here is the most introductory item I could find on them, but it's also pretty technical: esc.fnwi.uva.nl/thesis/centraal/files/f28012519.pdf – John Samples Apr 2 '16 at 0:54
• New way to confuse my engineer friends ... or better yet, the students working at the bio lab – Olórin Apr 2 '16 at 0:55
• See the answer at math.stackexchange.com/questions/488183/… – AHusain Apr 2 '16 at 22:13
• Sweet, thanks man! – John Samples Apr 2 '16 at 23:26