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3.4. NONLINEAR MULTIVARIATE EQUATIONS: RECURSIVE 99
3.4.2.2 Another Example
This was also considered as Example 1 (page 88).
Example 3 Let I = 〈ax 2 +x+y, bx+y〉 with the order a ≺ b ≺ y ≺ x. The full
traingular recomposition (obtained from Maple’s RegularChains package with
option=lazard) is
{
[bx + y, ay + b 2 − b], [x, y], [ax + 1, y, b], [x + y, a, b − 1] } .
The first two components correspond to two two-dimensional surfaces (in fact
planes), whereas the second two correspond to one-dimensional lines.
3.4.2.3 Regular Zeros and Saturated Ideals
Definition 64 If T is a triangular system, define the saturated ideal of T to
be
sat(T ) = {p ∈ K[x 1 , . . . , x n ]|∃n ∈ N init(T ) n p ∈ (T )}
= {p ∈ K[x 1 , . . . , x n ]|prem(p, T ) = 0} .
In other words it is the set of polynomials which can be reduced to zero by T
after multiplying by enough of the initials of T so that division works. In terms
of the more general concept of saturation I : S ∞ of an ideal ([Bou61, p. 90]),
this is (T ) : (init(T )) ∞ .
Theorem 23 ([ALMM99, Theorem 2.1]) For any non-empty triangular set
T ,
W (T ) = V (sat(T )).
In the example motivating Definition 63, sat(S) is generated by y−1, and indeed
W (S) = V (sat(S)).
3.4.3 Conclusion
Whether we follow the Gianni–Kalkbrener approach directly (algorithm 12) or
go via triangular sets, the solutions to a zero-dimensional family of polynomial
equations can be expressed as a union of (equiprojectable) sets, each of which
can be expressed as a generalised RootOf construct. For example, if we take the
ideal
{−3 x − 6 + x 2 − y 2 + 2 x 3 + x 4 , −x 3 + x 2 y + 2 x − 2 y, −6 + 2 x 2 − 2
y 2 + x 3 + y 2 x, −6 + 3 x 2 − xy − 2 y 2 + x 3 + y 3 },
its Gröbner basis (purely lexicographic, y > x) is
[6 − 3 x 2 − 2 x 3 + x 5 , −x 3 + x 2 y + 2 x − 2 y, 3 x + 6 − x 2 + y 2 − 2 x 3 − x 4 ]. (3.41)