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4.2. POLYNOMIALS IN TWO VARIABLES 125
Since this is a polynomial in y only, the greatest common divisor does not depend
on x, i.e. pp x (gcd(A, B)) = 1. Since cont x (A) = 1, cont x (gcd(A, B)) = 1, and
hence gcd(A, B) = 1, but we had to go via a polynomial of degree 10 to show
this. Space does not allow us to show bigger examples in full, but it can be
shown that, even using the subresultant algorithm (Algorithm 3), computing
the g.c.d. of polynomials of degree d in x and y can lead to intermediate 2
polynomials of degree O(d 2 ).
Suppose A and B both have degree d x in x and d y in y, with A = ∑ a i x i
and B = ∑ b i x i ). After the first division (which is in fact a scaled subtraction
C := a c B − b c A), the coefficients have, in general, degree 2d y in y. If we
assume that each division reduces the degree by 1, then the next result would
be λB − (µx + ν)C where µ = b c has degree d y in y and λ and ν have degree
3d y . This result has degree 5d y in y, but the subresultant algorithm will divide
through by b c , to leave a result D of degree d x − 2 in x and 4d y in y. Tthe next
result would be λ ′ C − (µ ′ x + ν ′ )D where µ ′ = lc x (C) has degree 2d y in y and λ ′
and ν ′ have degree 6d y . This result has degree 10d y in y, but the subresultant
algorithm will divide by a factor of degree 4, leaving an E of degree 6d y in y.
The next time round, we will have degree (after subresultant removal) 8d y in y,
and ultimately degree 2d x d y in y when it has degree 0 in x.
If this is not frigntening enough, consider what happens if, rather than being
in x and y, our polynomials were in n + 1 variables x and y 1 , . . . , y n . Then the
coefficients (with respect to x) of the initial polynomials would have degree d y ,
and hence (at most) (1+d y ) n , whereas the result would have (1+2d x d y ) n terms,
roughly (2d y ) n , or exponentially many, times as many terms as the inputs.
If we wish to consider taking greatest common divisors of polynomials in
several variables, we clearly have to do better. Fortunately there are indeed
better algorithms. The historically first such algorithm is based on Algorithm
14, except that evaluating a variable at a value replaces working modulo a prime.
Open Problem 5 (Alternative Route for Bivariate Polynomial g.c.d.)
Is there any reasonable hope of basing an algorithm on Algorithm 13? Intuition
says not, and that, just as Algorithm 14 is preferred to Algorithm 13 for the univariate
problem, so should it be here, but to the best of the author’s knowledge
the question has never been explored.
The reader will observe that the treatment here is very similar to that of the
univariate case, and may well ask “can the treatments be unified?” Indeed
they can, and this was done in [Lau82], but the unification requires rather more
algebraic machinery than we have at our disposal.
Notation 19 From now until section 4.3, we will consider the bivariate case,
gcd(A, B) with A, B ∈ R[x, y] ≡ R[y][x], and we will be considering evaluation
maps replacing y by v ∈ R, writing A y−v for the result of this evaluation.
2 We stress this word. Unlike the integer case, where the coefficients of a g.c.d. can be
larger than those of the original polynomials, the degree in y of the final g.c.d. cannot be
greater than the (minimum of) the degrees (in y) of the inputs.