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B.2. EQUALITY OF FACTORED POLYNOMIALS 223
with the f i and g j square-free and relatively prime, i.e.:w gcd(f i , f i ′) = 1,
gcd(g j , g j ′) = 1), is f ? =g.
The obvious solution is to expand f and g, and check that the expanded forms
(which are canonical) are equal. Can we do better?
One important preliminary remark is that the square-free representation of
a polynomial is unique. This leads to the following result.
Proposition 78 If f = g, then every a i has to be a b j , and vice versa. For
each such occurring value k, we must verify
f k =
n∏
?
f i =gk =
i=1
a i =k
m∏
g j .
j=1
b j =k
(B.1)
In particular, f k and g k must have the same degree, i.e.
n∑
deg(f i ) =
i=1
a i =k
m∑
deg(g j ).
j=1
b j =k
(B.2)
Again, we could check f k
?
=gk by expansion, but there is a better way.
Example 16 Let f = x 2l −1 and g = (x−1)(x+1)(x 2 +1) · · · (x 2l−1 +1), where
both are square-free, so proposition 78 does not help. f is already expanded, but
expansion of g can give rise to x 2l −1 + x 2l −2 + · · · + x + 1, which has length
O(2 l ), whereas g has length O(l).
From now on we will assume that we are working over a domain that includes
the integers.
Lemma 14 If f and g have degree at most N, and agree at N + 1 different
values, then f = g.
Proof. f − g is a polynomial of degree at most N, but has N + 1 zeros, which
contradicts proposition 5 if it is non-zero.
Hence it suffices to evaluate f and g at N + 1 points x i and check that the
values agree. It is not necessary to construct any polynomial, and the integers
involved are bounded (if we choose |x i | < N) by BN N , where B is a function
of the coefficients of the f i , g j . Furthermore, we can evaluate at all these points
in parallel. However, it does seem that we need to evaluate at N + 1 points, or
very nearly so, even if f and g are very small.
Open Problem 16 (Crossings of factored polynomials) Produce some nontrivial
bounds on the maximum number of zeros of f − g, where f and g have
small factored representations. See [RR90].
The best we can say is as follows. Suppose, in the notation of (B.1), each f i
has k i non-zero terms, and g j has l j non-zero terms, and no f i or g j is x (if