Do amoebas obtain extra tentacles as we take the tropical limit?

Question

Original Question

In this question, we'll restrict ourselves to plane curves.

Define the $t$-amoeba of a polynomial $p(z,w) = \sum_{i,j \in \mathbb{N}} a_{ij} z^i w^j$ to be the set $\mathcal{A}_t(p) = \{ (\log_t |z|, \log_t |w|) : (z,w) \in \mathbb{C}^2, p(z,w) = 0 \}$ (as defined in Gatthman's notes).

As we take the limit $\lim_{t \to \infty} \mathcal{A}_t(p)$, we expect a tropical curve to appear, with a degree $d$ polynomial having $d$ branches in the south, west, and northeast directions. For example, a degree 2 curve should degenerate to something like

Yet, when we look at pictures of Amoebas, say in the Wikipedia article, we don't see this regularity. For instance, in Wikipedia there's the example of the degree 3 polynomial

$P(z,w) = 3z^2+5zw+w^3+1$

having amoeba

In this case, which is $\mathcal{A}_e(P)$, we don't see the three tentacles in each standard direction. Coupled with that, there's this vacuole. Do the three tentacles we expect appear as we take the limit? What happens to the vacuole? Is there a specific $t$ when these tentacles appear and you start to see the tropical structure emerge in the spine?

Edit

I think these pictures might not have all the tentacles because the tentacles are overlapping. Is there a way to tell if a polynomial leads to a degenerate amoeba/tropical curve with some tentacles having multiplicity?

Expanded Answer

Thanks to Marco Golia's answer, I think I understand my misconception. I see now that we should expect any individual amoeba to converge to a tropical vertex - i.e., the tropical curve associated to a polynomial of form $$ \bigoplus_{i=1}^n z^{\odot a_z^{(i)}}\odot w^{\odot a_w^{(i)}} = \max \{ a_z^{(i)}z+a_w^{(i)}w : i=1,\dots,n \}$$ with $a_z^{(i)},a_w^{(i)} \in \mathbb{N}$. Every tropical plane curve can be viewed as a balanced graph, with vertices locally being of this form.

I've written code in Mathematica that allows you to generate these vertices/curves along with the convex hull associated to them here. I'll post the results for some of the Wikipedia amoebas below. Like Peter Taylor hinted in a comment, everything is just scaled down to the origin, and a lot of the tentacles collapse onto each other (as they get infinitely close to each other).

Amoeba

$P(z,w) = 3z^2+5zw+w^3+1$

Tropical Curve Associated

Amoeba

$P(z,w) = 50z^3 + 83z^2w + 24zw^2 + w^3 + 392z^2 + 414zw + 50w^2 - 28z + 59w - 100$

Tropical Curve Associated

Finally, for one last example

Amoeba

$P(z, w) = 1 + z + z^2 + z^3 + z^2w^3 + 10zw + 12z^2w + 10z^2w^2$

Tropical Curve Associated

Note

The image descriptions of the Tropical Curve should have the code input I used, and the Mathematica notebook should have sufficient description to what the code is outputting.

For more information/context, you can also view this short exposition I wrote.

Thanks!

Answer 1

This is not a complete answer and I'm not an expert, but it's way too long for a comment.

For a while I had the same misconception you have now, until Erwan Brugallé set me straight: there's no tropical curve associated to a complex curve. Rather, a tropical curve is associated to a degenerating family of complex curves. In your expression $\sum a_{ij}x^iy^j$, the $a_{ij}$ are supposed to be polynomials (or power series that converge around 0) in the variable $t$. Then you're taking the limit of a family of amoebas, and not a single amoeba. (Indeed, as Peter Taylor points out, if you did that this would degenerate to a tropical line with multiplicity.)

As for the way of detecting tentacles/vacoules/slopes/multiplicities, one gets a polygonal subdivision of the Newton polygon of the polynomial (the convex hull of the pairs $(i,j)$ for which $a_{ij} \neq 0$) which (if memory serves me well) comes from how fast $a_{ij}$ converges as $t\to 0$. The tropical curve in the limit is dual to this subdivision. This is much better explained in Brugallé and Shaw's notes here, in Section 5.

For the record, the vacuoles correspond to the genus of the complex curves in the family. For a non-singular tropical curve of degree $d$, the number of vacuoles that persist is $b_1$ of the corresponding graph (i.e. the number of cycles) and agrees with the genus of the corresponding complex curve (i.e. any of the non-singular complex curves in the family), which is $(d-1)(d-2)/2$. So, for a non-singular cubic, one vacuole, as in your example.