Polynomials [Lecture notes]

43 Pages·2007·0.31 MB·English

by  Grant Walker

MT30512: Polynomials Semester 2, 2006{07
Chapter 1: Polynomials and Ideals
1 Polynomials in n variables
1.1 Polynomials in one variable
A polynomialin the variablex with
oeÆ
ients inthe ring R is an expression
of the form
n
X
i n(cid:0)1 n
aix = a0 +a1x+:::+an(cid:0)1x +anx ; (1)
i=0
P
0
where n (cid:21) 0 and the ai are elements of R. If the notation is used, then x = 1
by
onvention. Note that x itself is NOT an element of R, but is a formal symbol
whi
h helps to make
al
ulations as natural as possible. If n = 0, the polynomial
is simply an element a0 2 R, and is
alled a
onstant polynomial.
The set of all polynomials (1) is a ring under the usual addition and multipli-

ation operations. It is denoted by R[x℄ and is
alled the ring of polynomials
in one variable over R. We shall use letters su
h as f;g;h or p;q;r to denote
polynomials; if it is ne
essary to draw attention to the variable being used, then
we write e.g. f(x), f(y) instead of f. We shall often assume that R is a (cid:12)eld F.
We shall always assume that R is a
ommutative ring with 1. You should
be familiar with the meaning of these terms from MT2262.
The most important fa
t about divisibility in F[x℄ is the division theorem.
If f 6= 0 (i.e. f is not the zero polynomial, so that some
oeÆ
ient ai 6= 0), then
we may assume in (1) that an 6= 0; n is then
alled the degree of the polynomial
f, and we write n = degf. Note that the degree of the zero polynomial is not
de(cid:12)ned.
Theorem 1.1 (Division Theorem) Let F be a (cid:12)eld and let f;g 2 F[x℄, where
g 6= 0. Then there are unique polynomials q, the quotient, and r, the remain-
der, in F[x℄ su
h that (i) f = qg+r, and (ii) either degr < degg or r = 0.
We shall see that the division theorem fails for polynomials in two or more
variables. Thisisonefundamentalreasonwhy thealgebraof polynomialsinmore
than one variable ismore diÆ
ult. One of the obje
ts of this
ourse is to
onvin
e
you that it is also more interesting and rewarding! However, before moving on,
we note two more things we
an do with one-variable polynomials.
(i) Substitution Given polynomials f(x);g(x) 2 R[x℄, we
an repla
e every
o

urren
e of x in f(x) by g(x) to obtain a new polynomial f(g(x)). For
2 2
example if f(x) = x +1 and g(x) = x(cid:0)2, then f(g(x)) = (x(cid:0)2) +1 =
2 2 2
x (cid:0)4x+5 and g(f(x)) = (x +1)(cid:0)2 = x (cid:0)1.
1
(ii) Evaluation Given a polynomial f(x) 2 R[x℄ and an element a 2 R, we

an repla
e every o

urren
e of x in f(x) by a to obtain a new element of
R, whi
h we denote by f(a). In this way, the polynomial f(x) de(cid:12)nes a
fun
tion from R to R. This fun
tion is
alled a polynomial fun
tion.
Although there is a logi
al distin
tionbetween polynomialsand polynomialfun
-
tions, it really does no harm to identify these two
on
epts. However, you should
be aware that two polynomials are equal if and only if they have identi
al
o-
eÆ
ients, whereas two fun
tions are equal if and only if they have the same
graphs. The di(cid:11)eren
e
an be seen by taking the (cid:12)eld F to be (cid:12)nite; for ex-
ample if p is a prime number and F = Fp, the (cid:12)eld with p elements, then the
p
polynomials x and x de(cid:12)ne the same fun
tion. But if F is an in(cid:12)nite (cid:12)eld su
h
as Q (the rational numbers), R (the real numbers) or C (the
omplex numbers),
two polynomials are equal if and only if the
orresponding polynomial fun
tions
are equal. Espe
ially for F = R, polynomial fun
tions give a geometri
way to
visualise polynomials via their graphs.
1.2 Polynomials in two variables
When we try to write down a de(cid:12)nition like (1) for a general polynomial in two
variables x and y with
oeÆ
ients in R, we have a diÆ
ulty: whi
h order do
we write the terms in? For example should the `standard form' of a quadrati
2 2
polynomial in x and y be a0;0+a1;0x+a0;1y+a2;0x +a1;1xy+a0;2y , or should
2
the term in x
ome before the term in y? Noti
e that we
an
olle
t terms in
2 2
either variable, and write this as (a0;0 +a1;0x+a2;0x )+(a0;1 +a1;1x)y +a0;2y
2 2
or as (a0;0 +a0;1y +a0;2y )+(a1;0 +a1;1y)x+a2;0x .
We
an hide this problem by writing the polynomial in the form
X
i j
ai;jx y (2)
i;j(cid:21)0
where ai;j 2 R and the sum is (cid:12)nite, i.e. ai;j = 0 for all but a (cid:12)nite number of
indexingpairs(i;j). Althoughtheproblemresurfa
es when we trytodopra
ti
al
i j

al
ulations, we adopt (2) as the formal de(cid:12)nition. We
all an expression x y a
i j
monomial in x and y, and if ai;j 6= 0 we
all ai;jx y a term of the polynomial.
The (total) degree of this monomial is i + j, and the (total) degree of a
nonzero polynomial is the maximum of the degrees of its terms. A polynomial is

alled homogeneous of degree d if it has no term of degree 6= d. (Thus the
zero polynomial is `homogeneous of degree d' for all d (cid:21) 0, although its degree is
not de(cid:12)ned!)
The set of allpolynomials(2) forms a ring R[x;y℄ with the usual addition and
multipli
ationoperations,
alledtheringofpolynomialsin two variables over
R. By
olle
ting terms with the same power of x, we
an regard a polynomial in
R[x;y℄ as a polynomial in x with
oeÆ
ients whi
h are polynomials in y, i.e. as
2
an element of S[x℄ where S = R[y℄. Thus we have a natural way to identify the
rings R[x;y℄ and (R[x℄)[y℄. Similarly, we
an identify R[x;y℄ and (R[y℄)[x℄.
As for the one variable
ase, we shall often use letters su
h as f;g;h or p;q;r
to denote polynomials; if it is ne
essary to draw attention to the variables being
used, thenwe writef(x;y)et
insteadoff. IfR = F isa(cid:12)eld,then apolynomial
f in F[x;y℄ is always divisible by any nonzero
onstant polynomial, i.e. any
polynomial of degree 0.
Itiseasytoseethatthedivisiontheoremfailsforpolynomialsintwovariables.
Suppose, for example, that we try to divide the polynomial x by the polynomial
y. This means we are looking for polynomials q and r su
h that x = qy +r and
degr < 1 or r = 0. Thus r must be a
onstant polynomial, and
learly there is
then no polynomial q 2 R[x;y℄ su
h that x = qy +r.
1.3 Polynomials in n variables
The de(cid:12)nitions above are easily extended to the
ase of n > 2 variables. A
monomial in n variables x1;:::;xn is an expression of the form
(cid:11)1 (cid:11)n
m = x1 (cid:1)(cid:1)(cid:1)xn ;
where (cid:11)1;:::(cid:11)n are integers (cid:21) 0. The (total) degree of the monomial m is
degm = (cid:11)1 + ::: + (cid:11)n. It is
onvenient to adopt a `multi-index' notation for
writing monomials, i.e.
(cid:11) (cid:11)1 (cid:11)n
x := x1 (cid:1)(cid:1)(cid:1)xn
where x = (x1;:::;xn) is a `ve
tor variable' and (cid:11) = ((cid:11)1;:::;(cid:11)n) is a `ve
tor
(cid:11)
exponent'. We denote the total degree of x by j(cid:11)j = (cid:11)1 +:::+(cid:11)n.
Let R be a
ommutative ring with 1. A polynomial over R in the variables
(cid:11)
x1;:::;xn is a (cid:12)nite sum f = f(x1;:::;xn) of terms of the form a(cid:11)x , where
(cid:11)
the
oeÆ
ient a(cid:11) 2 R. If a(cid:11) 6= 0, we say that the term a(cid:11)x appears in f. We
0 0
identify the monomial x1(cid:1)(cid:1)(cid:1)xn with the identity element 1 2 R. The
onstant
term of f is the element a(0;:::;0) 2 R; it may of
ourse be zero.
In multi-index notation, a polynomial
an be written as follows.
(cid:11)(1) (cid:11)(r)
f = a(cid:11)(1)x +:::+a(cid:11)(r)x (3)
where the
oeÆ
ients a(cid:11)(1);:::;a(cid:11)(r) areelements of R. The set of allpolynomials
of the form (3) forms a ring R[x1;:::;xn℄ with the usual addition and multipli
a-
tion operations,
alled the ring of polynomials in n variables over R. Noti
e
that by
olle
ting powers of one of the variables, xn say, we
an identify the rings
R[x1;:::;xn℄ and R[x1;:::;xn(cid:0)1℄[xn℄.
The (total) degree of a polynomial f 6= 0 is the maximum of the degrees of
the terms appearing in f . If all terms appearing in f have the same degree d,
f is
alled a homogeneous polynomial of degree d or a form of degree d.
In the
ase d = 1 we speak of a linear form, in the
ase d = 2 of a quadrati
form, in the
ase d = 3 of a
ubi
form, et
.
3
2 Ideals in polynomial rings
2.1 The ideal generated by a (cid:12)nite set
LetRbea
ommutativeringwith1andleta1;:::;an 2 R. Theideal generated
by a1;:::;an is the set of all elements of R of the form a = r1a1 + :::+ rnan,
where r1;:::;rn 2 R. We denote this ideal by ha1;:::;ani.
In parti
ular, the idealhai generated by a singleelement a 2 R is the set of all
multiples of a. In the general
ase, elements of the ideal are formed analogously
to linear
ombinations of ve
tors in a ve
tor spa
e.
2
Example 2.1 Let R = F[x;y℄ where F is any (cid:12)eld. Then hx ;yi is the set of
2
all polynomials f(x;y) of the form x g +yh where g and h are polynomials in
2
x and y. By thinking of f as a sum of monomials, we
an see that f 2 hx ;yi
2
if and only if every monomial whi
h appears in f is divisible by either x or y.
2
Thus hx ;yi is the set of all polynomials su
h that the
onstant term and the

oeÆ
ient of x are both zero.
2.2 Prin
ipal ideals
In general, an ideal in a ring R is a non-empty subset of R whi
h is
losed under
the operations of addition and multipli
ation by elements of R. More formally,
I (cid:18) R is an ideal if and only if
(cid:15) 0 2 I;
(cid:15) if a;b 2 I then a+b 2 I;
(cid:15) if a 2 I and r 2 R then ra 2 I.
It is easy to
he
k that if a1;:::;an 2 R then ha1;:::;ani is an ideal. Conversely,
given an ideal I (cid:18) R, we
an ask if there exist elements a1;:::;an 2 R su
h that
I = ha1;:::;ani. Su
h a set of elements is
alled a generating set or a basis
for I. If I is generated by a single element, i.e. if I = hai for some a 2 R, then I
is
alled a prin
ipal ideal of R.
2
It is easy to see from Example 2.1 that the ideal I = hx ;yi is not a prin
ipal
ideal in R = F[x;y℄. For if I = hfi, then the polynomial f would have to divide
2 2
both x and y. But the only
ommon divisors of x and y are nonzero
onstant
polynomials a 2 R. However a is invertible in R, and hen
e hai = R.
Remarkably, the situation is quite di(cid:11)erent in the
ase of polynomials in one
variable over a (cid:12)eld.
Theorem 2.2 Let F be a (cid:12)eld. Then every ideal in the polynomial ring F[x℄ is
prin
ipal.
4
Proof Let I (cid:18) F[x℄ be an ideal. If I = f0g then it is prin
ipal, so we may
assume that I
ontains a nonzero polynomial g of minimum degree. Clearly
hgi (cid:18) I. We shall prove that I (cid:18) hgi, so that I = hgi and hen
e I is prin
ipal.
Thus let f 2 I. By the Division Theorem 1.1, there exist polynomials q;r 2
F[x℄ su
h that f = qg + r and degr < degg or r = 0. Sin
e r = f (cid:0) qg and
f;g 2 I, it follows from the de(cid:12)nition of an ideal that r 2 I. By our
hoi
e of g,
it is impossible that degr < degg, and hen
e r = 0. But then f = qg, and so
f 2 hgi. (cid:3)
2.3 Finitely generated ideals
We see from Theorem 2.2 that the failure of the Division Theorem in the polyno-
mialringF[x;y℄intwo variablesover the(cid:12)eld F is
loselyrelated totheexisten
e
of ideals whi
h are not prin
ipal. From Example 2.1, we might be tempted to
guess that every ideal in F[x;y℄ is generated by at most two elements. The
following example shows that this is far too optimisti
.
n n(cid:0)1 n(cid:0)1 n
Example 2.3 Let F be a (cid:12)eld, let n (cid:21) 1 and let I = hx ;x y;:::;xy ;y i
be the ideal in F[x;y℄ generated by the set of allmonomialsof degree n. We shall
prove that I
annot be generated by fewer than n+1 polynomials.
First noti
e that the polynomials in I are the polynomials in whi
h no term
of degree < n appears. Let f1;:::;fm be a generating set for I: we must prove
i n(cid:0)i
that m (cid:21) n+1. Ea
h of the n+1 monomialsx y
an be expressed in the form
P
i n(cid:0)i m
x y = j=1gi;jfj where gi;j 2 F[x;y℄. Sin
e the fj have no terms of degree
P
i n(cid:0)i m
< n, equatingterms of degree n gives x y = j=1ai;jhj, where hj isthe degree
n homogeneous part of fj and ai;j 2 F is the
onstant term of gi;j. However, the
homogeneous polynomialsof degree n form a ve
tor spa
e of dimensionn+1 over
P
n n(cid:0)1 n(cid:0)1 n i n(cid:0)i m
F, with basis fx ;x y;:::;xy ;y g, and the equations x y = j=1ai;jhj
express ea
h of these basis elements as linear
ombinations of the h1;:::;hm.
Hen
e m (cid:21) n+1, by standard fa
ts of linear algebra.
WeseefromthisexamplethatthereisnointegerN su
hthateveryidealinF[x;y℄
is generated by some set with at most N elements. Obviously, the same is true
for F[x1;:::;xn℄ for every n (cid:21) 2. Thus the algebrai
stru
ture of a polynomial
ring in two or more variables is very di(cid:11)erent from the
ase of one variable. Here
are some questions we might ask about polynomials in n variables.
(cid:15) Dotwo polynomialshave agreatest
ommondivisor? (Re
allthat forn = 1
the answer is yes, and it
an be
omputed by the Eu
lidean Algorithm.)
(cid:15) Given an ideal I = hf1;:::fni and a polynomial f, how
an we determine
whether f is in I or not? (For n = 1, f 2 I if and only if the generator of
the ideal divides f.)
5
(cid:15) Does every ideal in F[x1;:::;xn℄ have a (cid:12)nite generating set?
The last question is answered by
Theorem 2.4 (Hilbert's Basis Theorem) Let F be a (cid:12)eld and let I be an ideal
in the polynomial ring P = F[x1;:::;xn℄. Then I is (cid:12)nitely generated, i.e. there
exists a positive integer N and a set of polynomials f1;:::;fN in I su
h that every
P
N
polynomial f 2 I
an be written in the form f = i=1rifi for some polynomials
r1;:::;rN in P.
We shall return to Hilbert's theorem later.
The following example shows that ideals in polynomial rings need not be
(cid:12)nitely generated in general. This example will not be needed in our future
work, and so the details (in
luding the pre
ise de(cid:12)nitions!) are left for you to
think about.
Exer
ise 2.5 Let F be a (cid:12)eld and let x1;x2;::: be an in(cid:12)nite sequen
e of vari-
ables. Let P be the polynomialring F[x1;x2;:::℄ and let I be the ideal
onsisting
of all polynomials with zero
onstant term. Then I is not (cid:12)nitely generated.
An ideal whi
h is not (cid:12)nitely generated
an always be generated by some set,
for example, the set of all its elements. Here we make pre
ise the de(cid:12)nition of a
generating set for an ideal.
Let R be a
ommutative ring with 1, let I be an ideal in R, and let A be a
subset of R. Then A is a generating set or basis for I (written I = hAi) if and
only if every element a 2 I
an be written in the form
a = r1a1 +:::+rnan; (4)
where a1;:::;an 2 A and r1;:::;rn 2 R, for some positive integer n.
The point is that although the set A
an be in(cid:12)nite, only (cid:12)nitely many ele-
ments of A are involved in (4) for a given a 2 I.
2.4 Quotient Rings
Re
all (MT2262) that when I is an ideal in the
ommutative ring R, then the
quotient ring R=I is
onstru
ted as follows. An element of P=I is a
oset
f = f +I = ff +g j g 2 Ig, and that the ring operations in P=I are de(cid:12)ned by
(f1 +I)+(f2 +I) = f1 +f2 +I; i:e: f1 +f2 = f1 +f2;
(f1 +I)(cid:1)(f2 +I) = f1 (cid:1)f2 +I; i:e: f1 (cid:1)f2 = f1 (cid:1)f2:
This is a good way to
onstru
t lots of interesting rings, as we shall see later.
6
3 Monomial Ideals and Di
kson's Lemma
Monomials are mu
h easier to handle than general polynomials. For example,
we
an write down the GCD and LCM of two given monomials at sight, e.g.
3 2 2 2 2 3 2 2 2 3 2 2
GCD(x y z;x z ) = x z, LCM(x y z;x z ) = x y z .
LetF bea(cid:12)eldandletP = F[x1;:::;xn℄bethepolynomialringinnvariables
over F. A monomial ideal in P is an ideal I whi
h is generated by a set of
monomials M. We do not assume that M is (cid:12)nite. Examples 2.1 and 2.3 are
examples of monomial ideals in F[x;y℄.
Theorem 3.1 Let I be a monomial ideal in P = F[x1;:::;xn℄, and let f be a
polynomial in P. Then f 2 I if and only if every monomial whi
h appears in f
is in I.
Proof The `if' part is obvious; we prove the `only if' part. Let f 2 I, and
(cid:11) (cid:11) (cid:11)
let a(cid:11)x be a term in f. Thus f = a(cid:11)x +f1, where the monomial x does not
(cid:11)
appear in f1. We must prove that x 2 I.
Let S be a set of monomials whi
h generates I. Then f is a (cid:12)nite sum
(cid:11)(1) (cid:11)(r)
f = g1x +:::+grx (5)
(cid:11)(1) (cid:11)(r)
for some polynomials g1;:::;gr 2 P and some monomials x ;:::;x 2 S.
Equating terms in (3.1) with exponent ve
tor (cid:11), we obtain
(cid:11) (cid:11)(1) (cid:11)(r)
a(cid:11)x = t1x +:::+trx (6)
where ea
h ti is either zero or is the term appearing in gi with exponent ve
tor
(cid:11)
(cid:12) = (cid:11)(cid:0)(cid:11)(i). Sin
e a(cid:11) 6= 0, equation (6) expresses x as an element of I. (cid:3)
This result redu
es the problem of testing a polynomial for membership of a
monomialideal I to the problem of testing a monomial. The next result provides
su
h a test.
(cid:11)
Theorem 3.2 Let S be a set of monomials in P = F[x1;:::;xn℄, and let x be
(cid:11) (cid:11)
any monomial in P. Then x 2 hSi if and only if x is divisible by some element
of S.
Proof Let I be the set of polynomialsf su
h that all monomials appearing
in f are divisible by some element of S. Then I is an ideal in P. Sin
e every
element of S is in I, we have hSi (cid:18) I. But if f 2 I, every monomial appearing
(cid:11)
in f is in hSi and so f 2 hSi. Hen
e I (cid:18) hSi, and so I = hSi. Taking f = x ,
we have the required statement. (cid:3)
Theorems 3.1 and 3.2 allow us to des
ribe the elements of a monomial ideal
3
hSi pre
isely. For example, hxy;y i (cid:26) F[x;y℄
onsists of all polynomials whi
h
2 n
have zero
onstant term and in whi
h the monomials y, y and all x (n (cid:21) 1) do
not appear.
7
3.1 Sums of ideals
Let I and J be ideals in R. Then their sum I +J is de(cid:12)ned by
I +J = fa+b j a 2 I;b 2 Jg: (7)
This is an ideal in R. If A and B are subsets of R su
h that A generates I and
B generates J, then I +J is generated by the union A [B of these generating
sets. It follows that the sum of two monomial ideals is again a monomial ideal.
2 3 4 2 2 3 4 2
Example 3.3 If I = hx y;y i and J = hx ;xy i, then I +J = hx y;y ;x ;xy i.
Noti
ethatingeneralI[J isnotanideal,butifI (cid:18) J thenI[J = J = I+J.
We
an think of I +J as the smallest ideal whi
h
ontains both I and J.
3.2 Produ
ts of ideals
Let I and J be ideals in R. Then their produ
t IJ is the set of elements r 2 R
of the form
n
X
r = aibi; where ai 2 I;bi 2 J: (8)
i=1
This is an ideal in R. If A and B are subsets of R su
h that A generates I and
B generates J, then IJ is generated by the produ
t AB = faibi j ai 2 A;bi 2 Bg
of these generating sets. It follows that the produ
t of two monomial ideals is
again a monomial ideal.
2 3 4 2 6 3 3 4 3 5
Example 3.4 If I = hx y;y i, J = hx ;xy i, then IJ = hx y;x y ;x y ;xy i.
4 3
Noti
e that we
an throw out x y from this list of generators, be
ause it is di-
3 3 6 3 3 5
visible by another monomialin the list, x y . The remaining list fx y;x y ;xy g
is a minimal generating set for IJ.
3.3 Interse
tions of ideals
The interse
tion I \J of ideals I and J in R is de(cid:12)ned to be their interse
tion
as sets, i.e. the set of all r 2 R su
h that r 2 I and r 2 J. This is an ideal in R.
(cid:11) (cid:12) (cid:11)
If x and x are monomials, then the interse
tion of the prin
ipal ideals hx i
(cid:12) (cid:13) (cid:13) (cid:11) (cid:12)
and hx i is the prin
ipal ideal hx i where x = LCM(x ;x ). For example,
2 3 2 3
hx yi \ hxy i = hx y i. This follows immediately from Theorem 3.2. More
generally, let I and J be the monomial ideals generated by sets of monomials S1
(cid:13)
and S2 respe
tively. Then by Theorem 3.2 a monomial x is in I \J if and only
(cid:13) (cid:11) (cid:12)
if x is divisible by some element x 2 S1 and also by some element x 2 S2.
(cid:11) (cid:12) (cid:11) (cid:12)
Hen
e the set fLCM(x ;x ) j x 2 S1;x 2 S2g is a set of monomial generators
for I \J.
8
2 3 4 2 4 2 2 4 3 3
Example 3.5 If I = hx y;y i and J = hx ;xy i, I \J = hx y;x y ;x y ;xy i.
4 3
Noti
ethatwe
anthrowouttheelementx y fromthelistofgenerators, be
ause
3 4 2 2 3
it is divisible by another monomial in the list, namely xy . Thus fx y;x y ;xy g
is a minimal generating set for I \J.
3.4 Di
kson's Lemma
The next result is a major step towards the proof of the Hilbert Basis Theorem.
Theorem 3.6 Let F be a (cid:12)eld, let S be a set of monomials in P = F[x1;:::;xn℄,
and let I = hSi. Then I is generated by a (cid:12)nite subset of the monomials in S.
In parti
ular, every monomial ideal in P is (cid:12)nitely generated.
Proof We (cid:12)rst prove that I isgenerated by some(cid:12)niteset ofmonomials,whi
h
need not belong to S. For this, we use indu
tion on n, the number of variables.
k
The
ase n = 1 is trivial, sin
e every monomial ideal is of the form hx i for some
k (cid:21) 0. Thus we assume that the result is true for n (cid:0) 1 variables, and let I
be a monomial ideal in P = F[x1;:::;xn℄. We shall write the variable xn as y,
(cid:11) m
so that monomials in P
an be written as x y , where x = (x1;:::;xn(cid:0)1) and
(cid:11) = ((cid:11)1;:::;(cid:11)n(cid:0)1).
Given a monomialideal I in P, let J be the ideal in F[x1;:::;xn(cid:0)1℄ generated
(cid:11) (cid:11) k
by the monomials x su
h that x y 2 I for some k (cid:21) 0. Then J is a monomial
ideal in F[x1;:::;xn(cid:0)1℄. By the indu
tion hypothesis, we
an
hoose a (cid:12)nite set
(cid:11)(1) (cid:11)(s)
of monomials fx ;:::;x g whi
h generates J.
(cid:11)(i) mi
For 1 (cid:20) i (cid:20) s, by de(cid:12)nition of J there is an integer mi su
h that x y 2 I.
Let m = maxmi. For 0 (cid:20) k (cid:20) m (cid:0) 1, let Jk be the ideal in F[x1;:::;xn(cid:0)1℄
(cid:11) (cid:11) k
generated by the monomials x su
h that x y 2 I. Then Jk is a monomialideal
in F[x1;:::;xn(cid:0)1℄, so again the indu
tion hypothesis allows us to
hoose a (cid:12)nite
set of these monomials fx(cid:11)k(1);:::;x(cid:11)k(sk)g whi
h generates Jk.
(cid:11)(i) m
We
laimthatI isgeneratedbythemonomialsx y ,for1 (cid:20) i (cid:20) s, together
with the monomials x(cid:11)k(i)yk, for 1 (cid:20) i (cid:20) sk and 0 (cid:20) k (cid:20) m(cid:0)1. This is a (cid:12)nite
set M of monomials.
(cid:11) k
Clearly all these monomials are in I. Let x y be a monomial in I. If k (cid:21) m,
(cid:11) (cid:11)(1) (cid:11)(s)
then sin
e x 2 J and J is the monomial ideal generated by fx ;:::;x g,
(cid:11) (cid:11)(i)
x is divisible by x for some i with 1 (cid:20) i (cid:20) s by Theorem 3.2. Hen
e
(cid:11) k (cid:11)(i) m
x y is divisible by x y . On the other hand, if 0 (cid:20) k (cid:20) m (cid:0) 1 then sin
e
x(cid:11) 2 Jk a similar argument proves that x(cid:11)yk is divisible by x(cid:11)k(i)yk for some i
with 1 (cid:20) i (cid:20) sk. Thus every monomial in I is divisible by some monomial in the
set M, and it follows that I = hMi.
To
omplete the proof, we must prove that a (cid:12)nite generating set for I
an
be sele
ted from any given set of monomials S whi
h generate the ideal I. Let
m1;:::;mr be a(cid:12)niteset ofmonomialswhi
h generatesI: su
h a set existsby the
(cid:12)rst part of the proof. By Theorem 3.2, ea
h mi is divisible by some monomial
0 0 0
mi 2 S, for 1 (cid:20) i (cid:20) r. Then m1;:::;mr lie in S and generate I. (cid:3)
9
4 Orderings on Monomials
We would like to have a division algorithm for polynomials in n variables. That
is, we would like to have an algorithm that has, as inputs, two polynomials f
and g and, as outputs, two polynomials q (the quotient of f by g) and r (the
remainder when f is divided by g). The polynomials q and r should satisfy
(1) f = qg+r, and
(2) r is `smaller' in some sense than g.
For this to work, we must
hoose an ordering on monomials. To see why,
2 2 2 2

onsider the example f = x y +xy , g = x y. Here it is natural to
hoose q = y
2 2 2
and r = xy . So we would like to regard xy as `smaller'than x y. However, if we
2 2 2 2
inter
hange the variables and divide x y +x y by xy , by symmetry we should
2 2
regard x y as the remainder and hen
e `smaller' than xy . To es
ape from this
2 2 2 2

ontradi
tion, we must (cid:12)rst agree whether to have x y > xy or x y < xy .
(cid:11)
De(cid:12)nition 4.1 Let F be a (cid:12)eld and let M be the set of all monomials x in
P = F[x1;:::;xn℄. A monomial ordering on P is a total ordering < on M
whi
h is
ompatible with multipli
ation, i.e. the following four axioms hold.
(cid:11) (cid:12)
(cid:15) (tri
hotomy rule) Given x ;x 2 M, exa
tly one of the three statements
(cid:11) (cid:12) (cid:12) (cid:11) (cid:11) (cid:12)
x < x , x < x and x = x is true.
(cid:11) (cid:12) (cid:13) (cid:11) (cid:12) (cid:12) (cid:13)
(cid:15) (transitive rule) If x ;x ;x 2 M satisfy x < x and x < x , then
(cid:11) (cid:13)
x < x .
(cid:11) (cid:11) (cid:11)
(cid:15) (initialisation rule) If x 2 M and x 6= 1, then 1 < x .
(cid:11) (cid:12) (cid:13) (cid:11) (cid:12) (cid:11) (cid:13) (cid:12) (cid:13)
(cid:15) (multipli
ation rule) If x ;x ;x 2 M and x < x , then x x < x x .
Along with the < symbol, we shall use the symbols (cid:20);>;(cid:21) with the obvious
meanings.
We give three examples. In these examples, we shall assume (as part of
the ordering) that the variables are ordered so that x1 > x2 > (cid:1)(cid:1)(cid:1) > xn. Let
(cid:11) (cid:11)1 (cid:11)n (cid:12) (cid:12)1 (cid:12)n
x = x1 (cid:1)(cid:1)(cid:1)xn , x = x1 (cid:1)(cid:1)(cid:1)xn .
(i) The lexi
ographi
ordering, Lex
(cid:11) (cid:12)
De(cid:12)ne x < x in Lex if and only if (cid:11)1 = (cid:12)1;:::;(cid:11)k(cid:0)1 = (cid:12)k(cid:0)1, (cid:11)k < (cid:12)k for
some k = 1;:::;n. In other words, reading from left (x1) to right (xn),
(cid:11)
the (cid:12)rst time the exponents are di(cid:11)erent the exponent in x is smaller.
(ii) The degree lexi
ographi
ordering, DegLex
(cid:11) (cid:12) (cid:11) (cid:12)
De(cid:12)ne x < x inDegLex ifandonlyifj(cid:11)j < j(cid:12)jorifj(cid:11)j = j(cid:12)j andx < x
in Lex. In other words, we use the degree (the sum of the exponents) as
the (cid:12)rst test, and use Lex to order monomials of the same degree.
10