Generic and Parallel Groebner Bases in JAS

We present generic, type safe Groebner bases software. The implemented algorithms distinguish Groebner base computation in polynomials rings over fields, rings with pseudo division, parameter rings, regular rings, Euclidean rings, non-commutative fields in commuting, solvable and free-non-commuting main variables. The interface, class organization is described in the object-oriented programming environment of the Java Algebra System (JAS). Different critical pair selection strategies and reduction algorithms can be provided by dependency injection. Different implementations can be selected for the mentioned coefficient rings through factory classes and methods. Groebner bases algorithms can be composed according to application needs and/or hardware availability. For example, versions for shared memory sequential or parallel computation, term order optimization or fraction free coefficient ring computation can be composed. For distributed memory compute clusters there are OpenMPI and MPJ implementations of Buchberger’s algorithm with optimized distributed storage of reduction polynomials.