
296 Algebraic Operads: An Algorithmic Companion
x
4
− x
2
2
− x
3
x
1
− x
2
x
1
− x
1
,
(2x
2
+ x
1
)(x
3
− x
2
− x
1
),
2x
3
1
+ 4x
2
2
+ 5x
3
x
1
− 3x
2
x
1
− 3x
2
1
− 2x
2
,
2x
2
x
2
1
− 7x
3
x
1
+ 7x
2
x
1
+ 9x
2
1
+ 2x
2
+ 2x
1
,
x
3
x
2
1
+ 2x
2
2
+ x
3
x
1
+ x
2
1
+ x
1
,
2x
2
2
x
1
− 2x
2
2
+ 5x
3
x
1
− 5x
2
x
1
− 7x
2
1
− 2x
1
,
x
2
3
x
1
+ x
2
2
+ x
3
x
1
+ x
2
1
+ x
2
+ x
1
,
2x
3
2
+ 2x
2
2
− x
3
x
1
+ x
2
x
1
+ x
2
1
− 2x
2
.
Examining the polynomial (2x
2
+ x
1
)(x
3
− x
2
− x
1
), we see that either
x
1
= −2x
2
or x
1
= x
3
− x
2
, so we again split into two cases.
Case 2.1: We substitute x
1
= −2x
2
in the last six polynomials of the set
above and compute the reduced Gröbner basis for the glex order, obtaining
{x
2
(x
2
− 1), x
2
(x
3
+ 2) }, for which the zero set is
[x
2
, x
3
] ∈ {[0, X] | X ∈ F } ∪ {[1, −2] ...