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4.2. POLYNOMIALS IN TWO VARIABLES 127
For the lemma to be false, the last g.c.d. has to be non-trivial. This implies
that the resultant Res x (A y−v /C y−v , B y−v /C y−v ) vanishes, by proposition 58 of
the Appendix. This resultant is the determinant of a Sylvester matrix 3 , and
|M y−v | = (|M|) y−v , for the determinant is only a sum of products of the coefficients.
In the present case, this amounts to saying that Res x (A/C, B/C) y−v
vanishes, that is that y − v divides Res x (A/C, B/C). But the hypotheses of the
lemma exclude this possibility.
Definition 71 If gcd(A y−v , B y−v ) = gcd(A, B) y−v , we say that the evaluation
of this problem at v is good, or that y − v is of good reduction. If not, we say
that y − v is of bad reduction.
This lemma implies, in particular, that there are only a finite number of values v
such that gcd(A y−v , B y−v ) does not have the same degree as that of gcd(A, B),
that is the y − v which divide the g.c.d. of the leading coefficients and the y − v
which divide the resultant of the lemma (the resultant is non-zero, and therefore
has only a finite number of divisors). In particular, if A and B are relatively
prime, we can always find a v such that A y−v and B y−v are relatively prime.
4.2.3 G.c.d. in Z p [x, y]
By Gauss’ Lemma (Theorem 5),
gcd(A, B) = gcd(cont x (A), cont x (B)) gcd(pp x (A), pp x (B)),
and the real problem is the second factor.
Observation 9 While the content of A = ∑ m
i=0 a ix i can be computed as
gcd(a m , gcd(a m−1 , gcd(a m−2 , . . . , a 0 ) . . .),
the following process is more efficient in practice (asymptotically, it’s only worth
a constant factor on what is asymptotically the cheaper operation, but in practice
it is worthwhile), and is valid over any g.c.d. domain R with a concept of ‘size’
equivalent to degree.
Algorithm 16 (Content)
Input: A = ∑ m
i=0 a ix i ∈ R[x].
Output: cont x (A)
S := {a i } # Really a set, as duplicates don’t matter
g := minimal degree element of S; S := S \ {g}
if g is a unit
3 There’s a subtle point here. The resultant of polynomials of degrees m and n is the
determinant of an (m + n) 2 matrix. Hence if y − v divides neither leading coefficient, the
Sylvester matrix of A y−v and B y−v is indeed the reduction of the Sylvester matrix of A and
B. If y −v divides one leading coefficient, but not the other, the Sylvester matrix of A y−v and
B y−v is smaller, and the reader should check that this only makes a difference of a product
of that leading coefficient which doesn’t vanish when v is substituted for y.