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2.1. WHAT ARE POLYNOMIALS? 35
not necessarily 10 irreducible. Hence this representation is generally known as
partially factored. In this format, the representation is not canonical, since the
polynomial x 2 −1 could be stored either as that (with 2 terms), or as (x−1)(x+1)
(with 4 terms): however, it is normal in the sense of definition 3. For equality
testing, see excursus B.2
Multiplication is relatively straight-forward, we check (via g.c.d. computations
11 ) for duplicated factors between the two multiplicands, and then combine
the multiplicands. Addition can be extremely expensive, and the result of an
addition can be exponentially larger than the inputs: consider
(x + 1)(x 2 + 1) · · · (x 2k + 1) + (x + 2)(x 2 + 2) · · · (x 2k + 2),
where the input has 4(k + 1) non-zero coefficients, and the output has 2 k+1
(somewhat larger) ones.
This representation is not much discussed in the general literature, but is
used in Redlog [DS97] and Qepcad [CH91], both of which implement cylindrical
algebraic decomposition (see section 3.5), where a great deal of use can be made
of corollaries 17 and 18.
2.1.4 Polynomials in several variables
Here the first choice is between factored and expanded. The arguments for, and
algorithms handling, factored polynomials are much the same 12 as in the case
of one variable. The individual factors of a factored form can be stored in any
of the ways described below for expanded polynomials, but recursive is more
common since it is more suited to g.c.d. computations (chapters 4 and 5), which
as we saw above are crucial to manipulating factored representations.
If we choose an expanded form, we have one further choice to make, which
we explain in the case of two variables, x and y, and illustrate the choices with
x 2 y + x 2 + xy 2 − xy + x − y 2 .
recursive — C[x][y]. We regard the polynomials as polynomials in y, whose
coefficients are polynomials in x. Then the sample polynomial would be
(x − 1)y 2 + (x 2 − x)y + (x 2 + x)y 0 . We have made the y 0 term explicit
here: in practice detailed representations in different systems differ on this
point.
10 If we were to insist on irreducibility, we would need to store x p −1 as (x−1)(x p−1 +· · ·+1),
with p + 2 terms rather than with 2. Furthermore, irreducibility can be expensive to prove
[ASZ00].
11 Again, these should be cheap if there is no factor to detect, and otherwise lead to reductions
in size.
12 In one variable, the space requirements for a typical dense polynomial and its factors are
comparable, e.g. 11 terms for a polynomial of degree 10, and 6 each for two factors of degree
5. For multivariates, this is no longer the case. Even for bivariates, we would have 121 terms
for a dense polynomial of degree 10, and 36 each for two factors of degree 5. The gain is
greater as the number of variables increases.