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3.4. NONLINEAR MULTIVARIATE EQUATIONS: RECURSIVE 97
3.4.2 Positive Dimension
Here we consider the case of solution sets of positive dimension (over the algebraic
closure, e.g. over the complexes). As in Theorem 22, the ultimate aim is
to express a variety as a union (preferably a disjoint union) of “nicer” varieties,
or other sets.
Definition 60 Quasi-algebraic System If P and Q are two (finite) sets of polynomials,
we call the ordered pair (P, Q) a quasi-algebraic system, and we write
Z(P, Q), the zeros of the quasi-algebraic system, for V (P ) \ V ( ∏ Q), with the
convention that if Q is empty, ∏ Q = 1, so V (Q) = ∅.
Z(P, Q) = {x ∈ K n |(∀p ∈ P p(x) = 0) ∧ (∀q ∈ Q q(x) ≠ 0)} .
We say that (P, Q) is consistent if Z(P, Q) ≠ ∅.
In Definition 43, we defined the variety of a set of polynomials, but we need some
more concepts, all of which depend on having fixed an order of the variables.
Definition 61 If T is a triangular system, we define the pseudo-remainder of
p by T to be the pseudo-remainder of dividing p by each q i ∈ T in turn (turn
defined by decreasing order of mvar(q i )), regarded as univariate polynomials in
mvar(q i ).
This is a generalization of Definition 30 (page 47).
Definition 62 Let S be a finite set of polynomials. The set of regular zeros of
S, written W (S), is Z(P, {init(P )}) = V (S) \ V ({init(S)}). For (a 1 , . . . , a n ) to
be in W (S), where S = {p 1 , . . . , p k }, we are insisting that all of the p i vanish
at this point, but none of the init(p i ).
For Example 2 of section 3.3.7, the variety is {(x = −1, y = 1), (x = 1, y = ±1)}.
If we take y > x, then the inital of the set of polynomials is lcm(1, x − 1, 1) =
x − 1, so only the zero with x = −1, y = 1 is regular. Conversely, if we take
x > y, the initial is y − 1 and only the zero with y = −1, x = 1 is regular. This
emphasises that W depends on the variable ordering. It is also a property of
the precise set S, not just the ideal 〈S〉.
In this case, W (S) was in fact a variety (as always happens in dimension 0).
In general, this is not guaranteed to happen: consider the (trivial) triangular
system S = {(x−1)y −x+1} with y > x. Since this polynomial is (x−1)(y −1),
V (S) is the two lines x = 1 and y = 1. However, W (S) is the line y = 1 except
for the point (1, 1). In fact this is the only direct description we can give,
though we could say that W (S) is “almost” the line y = 1. This “almost” is
made precise as follows.
Definition 63 If W is any subset of K n , the Zariski closure of W , written 16
W , is the smallest variety containing it:
W = ⋂ {V (F ) | W ⊆ V (F )},
which is itself a variety by Proposition 30.
16 Note that we use the same notation for algebraic closure and Zariski closure.