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76 CHAPTER 3. POLYNOMIAL EQUATIONS
Proposition 27 Every finitely generated polynomial ideal over a field K has a
completely reduced Gröbner base with respect to any given ordering, and this is
unique up to order of elements and multiplication by elements of K ∗ .
Hence, for a fixed ordering, a crGb is a “fingerprint” of an ideal, uniquely
identifying it. This makes definition 38 algorithmic. It also allows ideal arithmetic.
Proposition 28 Let G 1 and G 2 be Gröbner bases of the ideals I 1 and I 2 with
respect to a fixed ordering. Then:
1. I 1 ⊳ I 2 iff ∀g ∈ G 1 g ∗ → G2 0;
2. I 1 + I 2 = (G 1 ∪ G 2 );
3. I 1 I 2 = ({g 1 g 2 | g 1 ∈ G 1 , g 2 ∈ G 2 }) .
Furthermore, all these processes are algorithmic.
3.3.2 How many Solutions?
Here we will try to give an analysis of the various possibilities for the number
of solutions of a set of polynomial equations. We will assume that a crGb for
the polynomials has been computed, which therefore cannot be over-determined
in the sense of having redundant equations. However, we may still need more
equations than variables — see the examples at the start of section 3.3.7.
Unlike section 3.2.4 however, we have to ask ourselves “in which domain
are the solutions?” We saw in Theorem 8 that, even for an equation in one
variable, the ‘solutions’ may have no simpler formulation than ‘this is a root
of p(x)’. Fortunately, this is all that we need. We will assume that K is the
algebraic closure (definition 17) of (the field of fractions of) R.
Definition 43 The set of solutions over K of an ideal I is called the variety of
I, written V (I). If S is a set of polynomials which generates I, so I = 〈S〉, we
will write V (S) as shorthand for V (〈S〉).
We should note that two different ideals can have the same variety, e.g. (x) and
(x 2 ) both have the variety x = 0, but the solution has different multiplicity.
Proposition 29 V (I 1 · I 2 ) = V (I 1 ) ∪ V (I 2 ).
Proposition 30 V (I 1 ∪ I 2 ) = V (I 1 ) ∩ V (I 2 ).
Definition 44 The radical of an ideal I, denoted √ I, is defined as
√
I = {p|∀x ∈ V (I), p(x) = 0} .
If I is generated by a single polynomial p, √ I is generated by the square-free
part of p.