On Wikipedia there some pictures of two dimensional amoebas (Thanks to Oleg Alexandrov for the pictures and the Matlab code he gives to build them). I was wondering if somewhere there are pictures of 3-dimensional amoebas. Should it look likes thick foam bubbles?
You are welcome. The main reason for the difficulty of making such amoebas is that it is the projection of a 4-dimensional surface (the zero set of the polynomial). Finding a parametrization of the zeros set of an arbitrary polynomial in 2 variables is highly non-trivial. The case on wikipedia is linear, thus we may easily parametrize it. Now, this set is projected with $Log|\cdot|$, thus one has to be a bit careful which points to include to make the picture "pretty", since there may be zeros of the polynomial "far" away, and close to 0.
I did not draw the projection of the parametrized 4-dimensional surface, but rather the projection of a lot of points on the surface, chosen in a manner that makes them somewhat evenly distributed.
EDIT: Also, rumour has it that Frank Sottile has some 3-dimensional amoebas somewhere on his web page, but I have not been able to find them.
EDIT 2: Now, amoebas never look like soap bubbles, but the connected components of their complements do. (These components are always convex, aka. soap bubbles)