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2.5. THE BUCHBERGER ALGORITHM 19
Since g does not involve x, it holds that r 1 (2y 4 − 9) + r 2 (3x − 2y 3 )=−3s 2 . Thus,
(r 1 ,r 2 ) is a pair of polynomials for which r 1 (2y 4 − 9) + r 2 (3x − 2y 3 ) does not involve
x, but the degree of r 1 is strictly less than the degree of p 1 with respect to x,
yielding a contradiction unless r 1 ≡ 0. Then it follows that r 2 ≡ 0 and s 2 ≡ 0, so
that p 2 ≡ 0 and we have that g = s 1 (y)(2y 4 − 9). Clearly, for s 1 0, it follows that
LT (g) =2y 4 LT (s 1 ) and LT (g) ∈ LT (2y 4 , 3x).
(iii) To prove that I 1 = I 2 , consider the polynomial yf 1 − xf 2 =3x − 2y 3 = g 2 .
It holds that g 2 ∈ I 1 . Furthermore, −y 2 f 1 +(xy +3)f 2 = −yg 2 +3f 2 =2y 4 − 9=g 1
and it holds that g 1 ∈ I 1 . Therefore we conclude that I 2 ⊂ I 1 .
1
Conversely, it holds that:
3 g 1 + 1 3 yg 2 = f 2 and f 2 ∈ I 2 , and moreover that
2
9 y2 g 1 +( 2 9 y3 + 1 3 x)g 2 = 1 3 xg 2 + 2 3 y2 f 2 = f 1 and f 1 ∈ I 2 . From this we conclude that
I 1 ⊂ I 2 .
Combining I 2 ⊂ I 1 and I 1 ⊂ I 2 , yields I 1 = I 2 and {g 1 ,g 2 } is indeed a Gröbner
basis for I 1 .
When dividing by the elements of a Gröbner basis, the remainder will always be
unique, this in contrast to what holds for an arbitrary basis.
Proposition 2.18. Let G be a Gröbner basis for an ideal I in K[X] and let f be
a polynomial in K[X]. Then there is a unique r ∈ K[X] with the following two
properties:
(i) r is completely reduced with respect to G
(ii) there is h ∈ I such that f = h + r.
In particular, r is the remainder on division of f by G no matter how the elements
of G are listed when using the division algorithm.
For a proof see Proposition 1 of §2.6 of [28].
Proposition 2.18 can be used to determine if a polynomial belongs to an ideal or not;
also called the ‘ideal membership problem’: let G be a Gröbner basis for an ideal I
and f some polynomial in K[X]. Polynomial f belongs to ideal I if and only if the
remainder r on division of f by G is zero.
2.5 The Buchberger Algorithm
The Buchberger algorithm is a method of transforming a given set of generators for a
polynomial ideal into a Gröbner basis with respect to some monomial ordering. It was
developed by the Austrian mathematician Bruno Buchberger (see [23]). One can view
it as a generalization of the Euclidean algorithm for univariate GCD computation and
of Gaussian elimination for linear systems.
To transform a set of generators into a Gröbner basis, the S-polynomial is introduced.