link to my thesis

2.5. THE BUCHBERGER ALGORITHM 23
(ii) each p ∈ G is completely reduced modulo G −{p}.
Theorem 2.22. Every polynomial ideal I (not equal to {0}) has a unique reduced
Gröbner basis with respect to a monomial ordering.
For a proof see Proposition 6 of §2.7 of [28].
Example 2.12 (continued). A reduced Gröbner basis for the ideal I = 〈f 1 ,f 2 〉 =
〈x 2 − 2y 2 ,xy− 3〉 is G ′′ = {x − 2 3 y3 ,y 4 − 9 2 }.
The Buchberger algorithm yields a method to solve the ‘ideal equality problem’:
two ideals I 1 and I 2 have the same reduced Gröbner basis if and only if they are equal.
There are many improvements which have been made to this basic Buchberger
algorithm in the last decades, i.e., yielding the Refined Buchberger Algorithm or
the Faugère F4 Algorithm. Improvements are mostly aimed at avoiding redundancy
in the set of generators of the Gröbner basis or involving variations in efficiently
selecting a pair {p, q} ∈M. Other improvements are based on predicting a-priori
reductions of the S-polynomial which yield a zero solution. These reductions can then
be discarded. Despite all the improvements made to the basic Buchberger Algorithm
it is still computational very demanding because a Gröbner basis can be very large.
In fact, it can be much larger than the initial set of polynomials. For state-of-the-art
implementations of the Buchberger algorithm the reader may wish to consult Chapter
5.5 of [10] and [44].