Reduced Gröbner Basis

A reduced Gröbner basis is a Gröbner basis over a field in which every member is monic and no monomial appearing in a member is divisible by the leading monomial of a different member of . These conditions refer to a fixed monomial order. They apply to all terms, including the leading term.

Every ideal in a polynomial ring over a field has a unique reduced Gröbner basis for a fixed monomial order (Buchberger 2001b, p. 241). The basis is unique as a set, regardless of the order in which its members are listed. Thus two ideals are equal iff their reduced Gröbner bases for the same monomial order are equal. The zero ideal has the empty basis, and the whole polynomial ring has basis .

For example, for , the reduced Gröbner bases in the two lexicographic orders are

			(1)
			(2)

This illustrates why the monomial order must be fixed when asserting uniqueness. The basis for is a Gröbner basis but is not reduced because one member is not monic.

To obtain a reduced Gröbner basis from a Gröbner basis, repeatedly delete a member whenever the leading monomial of another remaining member divides its own. Stop when no such pair remains, keeping one representative if two leading monomials are equal. Make the remaining members monic and use polynomial reduction to reduce each member by the others.