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86 CHAPTER 3. POLYNOMIAL EQUATIONS
3.3.7 The Gianni–Kalkbrener Theorem
In this section, we will consider the case of dimension 0, i.e. finitely many
solutions over K. We first remark that the situation can be distinctly more
challenging than in the case of linear equations, which we illustrate by means
of two examples.
1. G = {x 2 − 1, y 2 − 1}. This is a Gröbner base with respect to any ordering.
There are four irreducible monomials {1, x, y, xy}, and hence four
solutions, x = ±1, y = ±1.
2. G = {x 2 − 1, y 2 − 1, (x − 1)(y − 1)}. This is also a Gröbner base with
respect to any ordering. There are three irreducible monomials {1, x, y},
and hence three solutions. There are x = 1, y = ±1, but when x = −1,
we only have y = 1. The additional polynomial (x − 1)(y − 1), which rules
out the monomial xy, rules out the solution x = y = −1. Another way of
looking at this is that, when x = 1, the polynomial (x−1)(y −1) vanishes,
but when x = −1, it adds an extra constraint.
Can we generalise this? The answer is ‘yes’, at least for purely lexicographical
Gröbner bases of zero-dimensional ideals. If the order is x n < x n−1 < · · · < x 1
then such a Gröbner base G must have the form
p n (x n )
p n−1,1 (x n−1 , x n ), . . . , p n−1,kn−1 (x n−1 , x n ),
p n−2,1 (x n−2 , x n−1 , x n ), . . . , p n−2,kn−2 (x n−2 , x n−1 , x n ),
· · ·
p 1,1 (x 1 , · · · , x n−1 , x n ), . . . , p 1,k1 (x 1 , · · · , x n−1 , x n ),
where deg xi
(p i,j ) < deg xi
(p i,j+1 ) and p i,ki is monic in x i . Let G k = G ∩
k[x k , . . . , x n ], i.e. those polynomials in x k , . . . , x n only.
Theorem 17 (Gianni–Kalkbrener [Gia89, Kal89]) Let α be a solution of
G k+1 . Then if lc xk (p k,i ) vanishes at α, then (p k,i ) vanishes at α. Furthermore,
the lowest degree (in x k ) polynomial of the p k,i not to vanish at α, say p k,mα ,
divides all of the other p k,j at α. Hence we can extend α to solutions of G k by
adding x k = RootOf(p k,mα ).
This gives us an algorithm to describe the solutions of a zero-dimensional ideal
from such a Gröbner base G. This is essentially a generalisation of backsubstitution
into triangularised linear equations, except that there may be more
than one solution, since the equations are non-linear, and possibly more than
one equation to substitute into.
Algorithm 9 (Gianni–Kalkbrener) GK(G, n)
Input: A Gröbner base G for a zero-dimensional ideal I in n variables with
respect to lexicographic order.
Output: A list of solutions of G.