Graphics Reference
In-Depth Information
fap ap
=
+
2 2
...
+
+
a
mm
11
for some polynomials a
i
(each of which is a sum of monomials), then f is sum of
products b
j
q
j
, where b
j
is a monomial and q
j
ΠP, and this provides a sequence of
elementary P-reductions that lead to 0. It follows that if the P-normal form is unique,
then all P-normal forms will be 0.
Proposition 10.10.7 motivates us to find bases P for ideals so that they induce unique
P-normal forms on polynomials. Such bases will be called Gröbner bases. There are
many possible equivalent definitions for these. We choose one based on leading term
properties rather than the unique P-normal form property because it is the former that
play the central role in all the proofs and practical algorithms.
Definition.
Fix a monomial order and let I be a nonzero ideal in k[X
1
,X
2
,...,X
n
].
A finite set P = {p
1
,p
2
,...,p
m
} of nonzero polynomials in I is called a
Gröbner basis
or
standard basis
for I if and only if for all nonzero f Œ I, there is some j, 1 £ j £ n, so
that lt(p
j
) divides lt(f).
Note that if we have a Gröbner basis P for an ideal I, then no nonzero polynomial
in I is P-irreducible.
Definition.
Let S be an arbitrary nonempty subset of k[X
1
,X
2
,...,X
n
]. Define the set
of leading terms of S, lt(S), by
()
=
{
()
}
.
1t S
lt f
f
Œ
S
10.10.8. Theorem.
Fix a monomial order and let I be a nonzero ideal in
k[X
1
,X
2
,...,X
n
]. The following statements are equivalent for a finite set P = (p
1
,p
2
,
...,p
m
) of nonzero polynomials in I:
(1) P is a Gröbner basis for I.
(2) f ΠI if and only if
ææ
f
0
.
(3) f ΠI if and only if
m
Â
()
=
{
() ()
}
f
=
a p
and
lpp f
max
lpp a
lpp p
,
ii
i
i
1
££
in
i
=
1
for some polynomials a
i
.
(4) <lt(P)>=<lt(I)>.
(5) Every polynomial f Πk[X
1
,X
2
,...,X
n
] has a unique P-normal form.
Proof.
We shall only prove that (1)-(4) are equivalent. For a proof showing that (5)
is equivalent to (2) see [AdaL94].
(1) fi (2): Let f ΠI. Let r = NF(f,P). Clearly, r ΠI. But r must be 0, otherwise one
would be able to reduce it further by the definition of a Gröbner basis. The converse
is also immediate.
