We give an alternative definition of comprehensive Gr ̈ obner bases in terms of Gr ̈ obner bases in polynomial rings over commutative Von Neumann regular rings. Our comprehensive Gr ̈ obner bases are defined as Gr ̈ obner 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 Gr ̈ obner bases. That is there exists a polynomial ideal generated by our comprehensive Gr ̈ obner basis which cannot be generated by any of Weispfenning’s original comprehensive Gr ̈ obner bases. © 2003 Elsevier Ltd. All rights reserved. MSC (1991):13B25; 13P10; 68W30
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