An alternative approach to comprehensive Gröbner bases

We give an alternative definition of comprehensive Gröbner bases in terms of Gröbner bases in polynomial rings over commutative Von Neumann regular rings. Our comprehensive Gröbner bases are defined as Gröbner bases in polynomial rings over certain commutative Von Neumann regular rings, hence they have two important properties which do not hold in standard comprehensive Gröbner bases. One is that they have canonical forms in a natural way. Another one is that we can define monomial reductions which are compatible with any instantiation. Our comprehensive Gröbner bases are wider than Weispfenning’s original comprehensive Gröbner bases. That is there exists a polynomial ideal generated by our comprehensive Gröbner basis which cannot be generated by any of Weispfenning’s original comprehensive Gröbner bases.