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72 CHAPTER 3. POLYNOMIAL EQUATIONS
not be zero, since then m i would be redundant). Then for each i, the monomial
version of equation (3.23) gives
P = Q & R = S implies a i m i P + b i m i R = a i m i Q + b i m i S.
Then we can use equation (3.15) repeatedly, with λ = 1, to add these together
to get the general form of equation (3.23).
Because of equation (3.2), we can regard equations as synonymous with
polynomials. Equation (3.23) then motivates the following definition.
Definition 37 Let S be a set of polynomials in the variables x 1 , . . . , x n , with
coefficients from R. The ideal generated by S, denoted (S), is the set of all
finite sums ∑ f i s i : s i ∈ S, f i ∈ R[x 1 , . . . , x n ]. If S generates I, we say that S
is a basis for I.
Proposition 21 This is indeed an ideal in the sense of definition 7.
Strictly speaking, what we have defined here is the left ideal: there are also
concepts of right ideal and two-sided ideal, but all concepts agree in the case of
commutative polynomials, which we will assume until section 3.3.12.
Proposition 22 ((S)) = (S).
Definition 38 Two sets of polynomial equations are equivalent if the polynomials
defining the left-hand sides generate the same ideal. We will see how to
test this in corollary 6.
Just as an upper triangular matrix is a nice formulation of a set of linear equations,
allowing us to “read off”, the solutions, so we would like a similarly ‘nice’
basis for an ideal generated by non-linear equations. In order to do this, we
will regard our polynomials in a distributed format, with the terms sorted in
some admissible (page 36) ordering >. Note that we are not requiring that
the polynomials are stored this way in an algebra system, though in fact most
algebra systems specialising in this area will do so: we are merely discussing
the mathematics of such polynomials. Having fixed such an ordering >, we can
define the following concepts.
Definition 39 If f is a non-zero polynomial, the leading term of f, denoted
lt(f), is that term greatest with respect to >. The corresponding monomial is
called the leading monomial of f, lm(f). We will sometimes apply lm to sets,
where lm(S) = {lm(s)|s ∈ S}.
“Monomial algebra” is a particularly simple form of polynomial algebra: in
particular
(
∏ n n
)
gcd x ai
i , ∏
n∏
x bi
i = x min(ai,bi)
i ,
lcm
i=1
( n
∏
i=1
x ai
i=1
i , n ∏
i=1
x bi
i
)
=
i=1
n∏
i=1
x max(ai,bi)
i .