Summary

Bertini Real is a real-solution-set decomposition method implemented using Bertini as its main computational engine.

The main algorithm used in Bertini Real is homotopy continuation:

generic homotopy continuation

In this method, we continue solutions from one system to another, through a homotopy.

Bertini Real uses as its input a text file, and the user passes the text file through two programs. First, the user performs a numerical irreducible decomposition using Bertini's tracktype:1, producing a file called 'witness_data'. Second, the user calls bertini_real on the same input file, with witness_data in place, and produces a real decomposition. This, of course, depends on the existence of a complex component of decomposable dimension – at this time, dimensions 1 and 2.

Bertini Real produces a folder of text files as output. Which files are produced depends on whether a curve or surface was decomposed. Common files are:

• the computed vertices,
• decomposition commonalities,
• witness set used to decompose,
• a copy of the input file,
• a copy of the witness data file.

Surfaces additionally produce:

• surface file, giving summary information,
• faces file, connecting computed vertices into 2-cells,
• all sub-decompositions, which are curves – critical curves, singular curves, the sphere intersection curve, and all mid- and crit-slice curves.

Curves produce:

• curve file, giving summary information,
• edges file, connecting computed vertices into 1-cells.