Following Zeb's suggestion (but a few second before he posted it) I came up with this imge

However, this "dequantization" procedure doesn't always produce the whole tropical curve. Here's the curve I drew to "requantize" Min[1, x , y , x+ y + 1, -2 + 2x , 2y]. A line has to be missing b/c of the zero tension condition, as in Tropical Mathematics by David Speyer and Bernd Sturmfels.

Here is the code. You have to draw 4 different versions of the curve to get all the absolute values. Maybe this should somehow involve complex phases as well.

q[x_] := E^( x)f[a_, b_, c_] := c + a + b + c a b + (1/c^2) a^2 + b^2;{x0, y0} = { -1, -3};L = 5;k = 8;    { f[q[k x], q[k y], q[k ]] == 0, f[-q[k x], q[k y], q[k ]] == 0,     f[q[k x], -q[k y], q[k ]] == 0, f[-q[k x], -q[k y], q[k ]] == 0},     {x, x0, x0 + L}, {y, y0, y0 + L}
        

Trying to draw the Amoeba

With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of 1 + x + y when k = 1. Then by plotting when $k \to \infty$, these graphs converge to the tropical polynomial Min(0,x,y).

However, I meant to draw Min(1,x,y) which I fixed by shifting my coordinates x,y by the vector (1,1). This is a bit ad-hoc and I am probably not understanding how constant functions "tropicalize".

Main question

I would like to "fatten" Min[x, y, 1, x + y + 1] into its amoeba, so I thought the right curve should be 1 + x + y + xy. However, my "neck" is disappearing in the scaling limit. How should I scale the coefficients correctly to get my amoeba?

Ideally, I want to take any tropical curve and fatten it into its amoeba. Tropical conics and cubics seem the best starting point.

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In anticipation of comments, by "amoeba" here I think I mean the boundary of the 2D region which is usually called "amoeba".

Trying to draw the Amoeba

With Mathematica, it's possible to graph $e^{-k x} + e^{-k y} = 1,e^{-k x} - e^{-k y} = 1$ and $e^{-k x} + e^{-k y} = -1$ to get the amoeba of 1 + x + y when k = 1. Then by plotting when $k \to \infty$, these graphs converge to the tropical polynomial Min(0,x,y).

However, I meant to draw Min(1,x,y) which I fixed by shifting my coordinates x,y by the vector (1,1). This is a bit ad-hoc and I am probably not understanding how constant functions "tropicalize".

Main question

I would like to "fatten" Min[x, y, 1, x + y + 1] into its amoeba, so I thought the right curve should be 1 + x + y + xy. However, my "neck" is disappearing in the scaling limit. How should I scale the coefficients correctly to get my amoeba?

Ideally, I want to take any tropical curve and fatten it into its amoeba. Tropical conics and cubics seem the best starting point.

---

In anticipation of comments, by "amoeba" here I think I mean the boundary of the 2D region which is usually called "amoeba".