Discrete differential geometry is only a few years old but has already provided much insight into fundamental aspects of classical differential geometry and of the theory of integrable systems (see A.I. Bobenko, Yu.B. Suris (2005) math.DG/0504358).
In the center of attention are so-called `face' relations between weights associated to corners and edges of an n-dimensional quadrilateral with the condition that these quadrilaterals allow the consistent assembly of a (n+1)-dimensional discrete net.
In the talk a classification of symmetric 3-dimensional scalar `face relations' (discrete integrable equations) is presented that are 4-dimensional consistent. This work was done in collaboration with Sergey Tsarev, following suggestions of Alexander Bobenko.
The talk starts with a description of the computational problem followed by a brief description of probabilistic methods that had to be introduced in order to handle the otherwise astronomically large polynomial algebraic system.
Being written in REDUCE the computer algebra package CRACK running under Parallel REDUCE allows a parallel investigation of different cases of the non-linear problem. The talk expands in more detail on guidelines that were followed in parallelizing the computation and constraints that appear when trying to use large computer clusters that had been designed to run numerical computations in batch mode instead of symbolic computations in a semi-interactive mode.