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134 CHAPTER 4. MODULAR METHODS
(presumed) x i y j terms in the g.c.d., to get the following final result:
x 4 (y 7 ( z 3 + z 2 + z + 1 ) + y 3 ( z 4 + z 3 + z 2 + z + 1 ) +
y ( z 5 + z 4 + z 3 + z 2 + z + 1 )) + x 2 ( y 7 ( z 3 + z 2 + z + 1 ) + y 4 z 5 + 1 ) + 1.
(4.13)
We still need to check that it does divide g and g, but, once we have done so,
we are assured by Corollary 11 that it is the greatest common divisor.
4.3.2 Converting this to an algorithm
There are, however, several obstacles to converting this to an algorithm: most
centring around the linear systems such as (4.10). First, however, we must
formalize the assumption made on page 133.
Definition 72 We define the skeleton of a polynomial f ∈ R[x 1 , . . . , x n ], denoted
Sk(f), to be the set of monomials in a (notional) distributed representation
of f.
Definition 73 We say that an evaluation φ : R[x 1 , . . . , x n ] → R[x i : i /∈ S],
which associates a value in R to every x i : i ∈ S, satisfies the Zippel assumption
for f if ψ(Sk(f)) = Sk(φ(f)), where ψ deletes the powers of x i : i /∈ S from
monomials.
For example, if f = (y 2 − 1)x 2 − x, Sk(f) = {x 2 y 2 , x 2 , x}. If φ : y ↦→ 1, then
φ(f) = −x, so Sk(φ(f)) = {x}, but ψ(Sk(f)) = {x 2 , x}, so this φ does not
satisfy the Zippel assumption. However, if φ : y ↦→ 2, φ(f) = 3x 2 − x and
Sk(φ(f)) = {x 2 , x}, so this φ satisfies the Zippel assumption.
Hence the assumption on page 133 was that z ↦→ 2 satsifies the Zippel
assumption. Let us now look at the problems surrounding (4.10). In fact, there
are three of them.
1. The equations might be under-determined, as we saw there.
2. We need to solve a t × t system, where t is the number of terms being
interpolated. This will normally take O(t 3 ) coefficient operations, and, as
we know from section 3.2.3, the coefficients may also grow during these
operations.
3. The third problem really goes back to (4.9). We stated that
gcd(f| z=3,y=−1 , g| z=3,y=−1 ) = −525x 4 + 204x 2 + 1,
but we could equally well have said that it was 525x 4 − 204x 2 − 1, at
which point we would have deduced that the equivalent of (4.10) was
inconsistent. In fact, Definition 25 merely defines a greatest common
divisor, for the reasons outlined there. We therefore, as in Algorithm 14,
impose gcd(lc(f), lc(g)), as the leading coefficient of the greatest common
divisor.