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92 CHAPTER 3. POLYNOMIAL EQUATIONS
• each primary component is simple or of local dimension 1.
Furthermore ([BMMT94, ) Lemma 2], such a rotation need only be 1-generic, i.e.
1 v
have matrix for some generic vector v.
0 I
Their paper generalises this to ideals of higher dimension, but the complexity
in notation is not worth it for our purposes.
A word of warning is in order here. The Shape Lemma is a powerful theoretical
tool, but its application can be costly. Consider example 2 (page 86):
G = {x 2 − 1, y 2 − 1, (x − 1)(y − 1)}. This is certainly not a shape basis, since it
has more polynomials than indeterminates. This is inevitable, since the variety
is not equiprojectable (see Definition 58 below) onto either x or y. If we write
s = x + y, t = x − y, then the basis becomes {−4 t + t 3 , −4 + t 2 + 2 s} for the
ordering 13 s > t, which is a shape basis. However, consider the similar basis
G ′ = {x 2n −1, y 2n −1, (x n −1)(y n −1)}. Similar rotations will work, but t is now
the root of a polynomial of degree 3n 2 with at least 3n + 1 nonzero coefficients,
and quite large ones at that, e.g. for n = 3
{
8469703983104 − 1328571568128 t 3 + 56109155544 t 6 − 3387236203 t 9
+149161506 t 12 − 11557977 t 15 + 279604 t 18 − 1053 t 21 − 78 t 24 + t 27 ,
−586877251095044672 t + 11229793345003520 t 4 − 363020550569195 t 7
+24557528419410 t 10 − 3328382464425 t 13 + 88786830300 t 16 − 417476125 t 19
−23303630 t 22 + 307217 t 25 + 259287804304663680 s } .
3.3.10 The Hilbert function
Let I be any ideal (other than the whole ring) of k[x 1 , . . . , x n ]. Let A be the
algebra k[x 1 , . . . , x n ]/I, i.e. the set of all polynomials under the equivalence
relation f ≡ g if there is an h ∈ I with f = g + h.
Proposition 36 If G is a Gröbner base for I, then A is generated, as a k-
vector space, by M := {m monomial ∈ k[x 1 , . . . , x n ]|m ∗ → G m}, i.e. the set of
irreducible monomials.
We have already seen (Proposition 32) that the variety corresponding to I is
zero-dimensional if, and only if, M is finite. In this case, |M| is the number of
solutions (counted with multiplicity).
The Hilbert function is a way of measuring M when it is infinite.
Definition 52 Let A l be the subset of A where there is a representative polynomial
of total degree ≤ l. A l is a finite-dimension vector space over k. Let H I
be the function N → N defined by
H I (l) = dim k A l . (3.34)
13 But not for t > s, since s defines the other line, besides the x and y axes, going through
two of the points.