An alternative approach to comprehensive Gro¨bner bases

We give an alternative definition of comprehensive Gro¨bner bases in terms of Gro¨bner bases in polynomial rings over commutative Von Neumann regular rings. Our comprehensive Gro¨bner bases are defined as Gro¨bner bases in polynomial rings over certain commutative Von Neumann regular rings, hence they have two important properties which do not hold in standard comprehensive Gro¨bner 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 Gro¨bner bases are wider than Weispfenning’s original comprehensive Gro¨bner bases. That is there exists a polynomial ideal generated by our comprehensive Gro¨bner basis which cannot be generated by any of Weispfenning’s original comprehensive Gro¨bner bases. [Copyright &y& Elsevier]