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2.5. THE BUCHBERGER ALGORITHM 21
The algorithm above gives a set of generators G = {g 1 ,...,g t } which is a Gröbner
basis for the ideal I. If there is any member of G whose leading term can be divided
by the leading term of any other member of G, one can eliminate this member from
the basis to get a minimal size of the basis.
Example 2.12 (continued). Suppose we start with the set of polynomials {f 1 ,f 2 } =
{x 2 − 2y 2 ,xy − 3} of Example 2.12 where the monomials are ordered with respect
to a lexicographic ordering. In Example 2.12 the Gröbner basis for I = 〈f 1 ,f 2 〉 is
given by {g 1 ,g 2 } = {2y 4 − 9, 3x − 2y 3 }. We now show step-by-step how to obtain this
Gröbner basis for the ideal I using the Buchberger algorithm as shown in Algorithm
1 (each line starts with a number denoting the line of pseudocode in Algorithm 1).
1 The set G is initialized as G = F = {f 1 ,f 2 }.
2 The set M is initialized as M = {{f i ,f j } : f i ,f j ∈ G, f i f j } = {{f 1 ,f 2 }}.
Note that {f 2 ,f 1 } need not be included.
3 The set M is not empty: new iteration starts.
4 The pair {p, q} can only be chosen as {f 1 ,f 2 }.
5 The set M becomes: M = M −{f 1 ,f 2 } = ∅.
6 The S-polynomial of {f 1 ,f 2 } is computed using LT (f 1 )=x 2 , LT (f 2 )=xy and
J = LCM(LM(f 1 ),LM(f 2 )) = LCM(x 2 ,xy)=x 2 y as: S = S-polynomial(f 1 ,
f 2 )=
J
LT (f f 1) 1 −
J
LT (f f 2) 2 = x2 y
x
f 2 1 − x2 y
xy f 2 = yf 1 − xf 2 =3x − 2y 3 = g 2 .
7 The NormalForm of g 2 is computed as h = NormalForm(g 2 ,G)=g 2 =3x−2y 3
because:
3x is not divisible by LM(f 1 )=x 2 nor divisible by LM(f 2 )=xy.
−2y 3 is not divisible by LM(f 1 )=x 2 nor divisible by LM(f 2 )=xy.
Therefore the polynomial g 2 is already in normal form.
8 Because h = g 2 0, the sets M and G are extended.
9 The set M is extended to M = {{f 1 ,g 2 }, {f 2 ,g 2 }}.
10 The set G becomes G = {f 1 ,f 2 ,g 2 } = {x 2 − 2y 2 ,xy− 3, 3x − 2y 3 }.
3 The set M is not empty: new iteration starts.
4 The pair {p, q} is chosen as {f 1 ,g 2 }.
5 The set M becomes: M = M −{f 1 ,g 2 } = {{f 2 ,g 2 }}.
6 The S-polynomial of {f 1 ,g 2 } is computed using LT (f 1 )=x 2 , LT (g 2 )=3x and
J = LCM(LM(f 1 ),LM(g 2 )) = LCM(x 2 ,x)=x 2 as: S = S-polynomial(f 1 ,
g 2 )=
J
LT (f 1) f 1 −
2
3 xy3 − 2y 2 .
2
7 The NormalForm of
J
LT (g 2) g 2 = x2
x 2 f 1 − x2
3x (3x − 2y3 )=x 2 − 2y 2 − x 2 + 2 3 xy3 =
3 xy3 − 2y 2 is computed as h = NormalForm( 2 3 xy3 −
2y 2 ,G)= 2 3 xy3 − 2y 2 − 2 3 (xy − 3)y2 =0.
8 h =0.
9 Because h = 0 the set M is not extended.
10 Because h = 0 the set G is not extended.
3 The set M is not empty: new iteration starts.
4 The pair {p, q} is chosen as {f 2 ,g 2 }.
5 The set M becomes: M = M −{f 2 ,g 2 } = ∅.