This paper on the Arxiv is a nice intro to Tropical Geometry:
http://arxiv.org/pdf/math/0601322v1.pdf

Calling Matlab from C++ or C is kinda a pain for me right now. Bertini_real uses Matlab for its symbolic engine, something that I dislike at any rate, and my current interface is through a system(); call. Ugh.

Today, I just got a C++ program to compile against Matlab's libraries, and the program opens the engine and closes it. A good first task.

My compiling command was:
clang -I /Applications/MATLAB_R2014a.app/extern/include/ matlabtest.cpp -L/Applications/MATLAB_R2014a.app/bin/maci64 -leng -lm -lmat -lmex -lut -lmx -lc++
on my 10.10 macintosh. In order to run the output application a.out, I had to add the Matlab library paths to my runtime library path:
export DYLD_LIBRARY_PATH=$DYLD_LIBRARY_PATH:/Applications/MATLAB_R2014a.app/bin/maci64/
which is something that I dislike, as I have no idea what other libraries are in that path, and which ones may mash necessary libraries for other applications. I will continue to investigate how to NOT have to set DYLD_LIBRARY_PATH.

One outstanding potential issue -- will I be able to link against Windows Matlab libraries and headers from within Cygwin???