A direct products of fields approach to comprehensive Grobner bases over finite fields

In this paper we describe comprehensive Grobner bases over finite fields by direct product of fields. In general, representations of comprehensive Grobner bases have some conditions on parameters. However, in finite fields we can construct comprehensive Grobner bases without conditions by the theory of von Neumann regular rings. Our comprehensive Grobner bases are defined as Grobner bases in polynomial rings over commutative von Neumann regular rings, hence our comprehensive Grobner bases have some nice properties. Our method is different from the methods of Weispfenning (CGB,CCGB), Monies (DisPGB), Sato and Suzuki (ACGB).