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144 CHAPTER 4. MODULAR METHODS
4.5.2 The Hilbert Function and reduction
We recall the Hilbert function from section 3.3.10, which will turn out to be
a key tool in comparing Gröbner bases, as earlier we have used degrees for
comparing polynomials.
Theorem 29 ([Arn03, Theorem 5.3]) Let (f 1 , . . . , f k ) generate the ideal I
over Q, and the ideal I p moulo p. Then
∀n ∈ N : H I (n) ≤ H Ip (n). (4.20)
Definition 75 p is said to be Hilbert-good if, and only if, we have equality in
(4.20).
Observation 11 Note that we do not have a test for Hilbert-goodness as such,
but we do have one for Hilbert-badness: If H Ip (n) < H Iq (n) then q is definitely
Hilbert-bad.
Open Problem 7 (Contradictory Hilbert Functions) It is conceivable to
have H Ip (n) < H Iq (n) but H Ip (m) > H Iq (m), in which case both p and q must
be bad. Can we give any examples of this?
Proposition 47 If p is of good reduction for C(S), then it is Hilbert-good for
the ideal generated by S.
Theorem 30 ([Arn03, Theorem 5.6]) Let (g 1 , . . . , g t ) be a Gröbner base under
< for the ideal generated by (f 1 , . . . , f s ) over Q, and (g 1, ′ . . . , g t ′ ′) be a
Gröbner base under < for the ideal generated by (f 1 , . . . , f s ) modulo p. In both
cases these bases are to be ordered by increasing order under x’, and consider the primes 5 and 2.
I 5 has Gröbner base
{
3 y 2 x + 2 x 3 , 29 yx 3 , x 5 , y 6} , (4.21)
whereas I 2 has Gröbner base
{
y 2 x + yx 2 , y 6 + yx 5} . (4.22)