Constructions in Finite Geometry Using Computer Algebra Systems

One way of using a computer algebra system to do research in finite geometry is to use the system to construct “small" order examples of various constructions, and then hope to recognize a pattern that can be generalized and eventually proven. Of course, initially one does not know if the “small" order examples exist. However, if one has sufficiently good insight concerning where to look and a reasonably good “starter", the computer algebra system will often find these examples quite expeditiously. Once found the system can then be used to analyze the constructs. Brute-force searching, on the other hand, is typically foolhardy with such general purpose systems. These ideas will be illustrated with two problems in finite geometry: (1) finding new translation planes by a technique called “nesting", and (2) finding large collections of pairwise disjoint projective bundles of conics.