
258 Algebraic Operads: An Algorithmic Companion
From the first two elements we see that DI
2
(A) contains x
1
, and then
the fourth element shows that DI
2
(A) contains x
2
. Since none of these
elements has a nonzero constant term, we conclude that [x
1
, x
2
] is a
Gröbner basis for DI
2
(A). It follows that DI
2
(A) = (x
1
, x
2
) and hence
V (DI
2
(A)) = {(0, 0)}. This implies that rank(A) < 2 if and only if
x
1
= x
2
= 0. Since we already know that rank(A) ≥ 1, we conclude that
the inverse image of r = 1 is the single point {(0, 0)}.
For r = 3, there is only one determinant to compute: det(A) = x
3
1
− x
2
1
,
and this polynomial is a Gröbner basis for the (principal) ideal D