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4.1. GCD IN ONE VARIABLE 115
The concept of modular methods is inspired by this calculation, where there
is no possibility of intermediate expression swell, for the integers modulo 5 are
bounded (by 4). Obviously, there is no need to use the integers modulo 5: any
prime number p will suffice (we chose 5 because the calculation does not work
modulo 2, for reasons to be described later, and 3 divides one of the leading
coefficients). In this example, the result was that the polynomials are relatively
prime. This raises several questions about generalising this calculation to an
algorithm capable of calculating the g.c.d. of any pair of polynomials:
1. how do we calculate a non-trivial g.c.d.?
2. what do we do if the modular g.c.d. is not the modular image of the g.c.d.
(as in the example in the footnote 1 )?
3. how much does this method cost?
4.1.1 Bounds on divisors
Before we can answer these questions, we have to be able to bound the coefficients
of the g.c.d. of two polynomials.
Theorem 27 (Landau–Mignotte Inequality [Lan05, Mig74, Mig82]) Let
Q = ∑ q
i=0 b ix i be a divisor of the polynomial P = ∑ p
i=0 a ix i (where a i and b i
are integers). Then
q
max
i=0 |b i| ≤
√
q∑
∣ ∣ √√√ ∣∣∣
|b i | ≤ 2 q b q ∣∣∣
p∑
a 2 i
a .
p
i=0
These results are corollaries of statements in Appendix A.2.2.
If we regard P as known and Q as unknown, this formulation does not
quite tell us about the unknowns in terms of the knowns, since there is some
dependence on Q on the right, but we can use a weaker form:
√ √√√ q∑
p∑
|b i | ≤ 2 p a 2 i .
i=0
When it comes to greatest common divisors, we have the following result.
Corollary 8 Every coefficient of the g.c.d. of A = ∑ α
i=0 a ix i and B = ∑ β
i=0 b ix i
(with a i and b i integers) is bounded by
i=0
i=0
⎛
2 min(α,β) gcd(a α , b β ) min ⎝ 1
√ α ∑
a 2 i
|a α |
, 1
∑
√ β
|b β |
i=0
i=0
b 2 i
⎞
⎠ .