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52 CHAPTER 2. POLYNOMIALS
6) to be whichever of x and y minimises
min(max(deg x (f), deg x (g)), max(deg y (f), deg y (g))).
v = 3 Here the coefficients are bivariate polynomials. If we assume classic multiplication
on dense polynomials, F (d) = cd 4 +O(d 3 ). We are then looking
at
k∑
(k − i)F (id) ≤ c
i=0
≤
ck
k∑
(k − i)i 4 d 4 2 +
i=0
k∑
i 4 d 4 − c
i=0
k∑
kO(i 3 d 3 )
i=0
k∑
i 5 d 4 + k 5 O(d 3 )
i=0
( ) ( )
1 1
= c
5 k6 + · · · d 2 − c
6 k6 + · · · d 2 + k 5 O(d 3 )
( ) 1
= c
30 k6 + · · · d 4 + k 5 O(d 3 )
which we can write as O(k 6 d 4 ). The asymmetry is again obvious.
general v The same analysis produces O(k 2v d 2v−2 ).
We see that the cost is exponential in v, even though it is polynomial in d and
k. This is not a purely theoretical observation: any experiment with several
variables will bear this out, even when the inputs (being sparse) are quite small:
the reader need merely use his favourite algebra system on
a 0 := ax 4 + bx 3 + cx 2 + dx + e; a 1 := fx 4 + gx 3 + hx 2 + ix + j,
treating x as the main variable (which of course one would not do in practice),
to see the enormous growth of the coefficients involved.
2.3.5 Square-free decomposition
Let us revert to the case of polynomials in one variable, x, over a field K, and let
us assume that char(K) = 0 (see definition 15 — the case of characteristic nonzero
is more complicated [DT81], and we really ought to talk about ‘separable
decomposition’ [Lec08]).
Definition 31 The formal derivative of f(x) = ∑ n
i=0 a ix i is written f ′ (x) and
computed as f ′ (x) = ∑ n
i=1 ia ix i−1 .
This is what is usually referred to as the derivative of a polynomial in calculus
texts, but we are making no appeal to the theory of differentiation here: merely
defining a new polynomial whose coefficients are the old ones (except that a 0
disappears) multiplied by the exponents, and where the exponents are decreased
by 1.