Biology Reference
In-Depth Information
The next lines compute each
f
“modulo”
I
and is the same as
NF
(
f
,
G
)
, where
G
is the reduced Gröbner basis for
I
with respect to the monomial order given in the
description of the polynomial ring.
f1%I
f2%I
f3%I
(
f1
,
G
)
=−
x
3
−
,
(
f2
,
G
)
=
x
2
−
(
f3
,
G
)
=
We see that
NF
1
NF
2, and
NF
−
is the Gröbner basis for
I
with respect to
grevlex
.
Now let's change the order to
lex
. Since the
f
polynomials and
I
were defined as
objects in the ring
R
, we need to “tell” Macaulay 2 to consider them as objects in a
new ring with a different monomial order.
2
x
3
+
1, where
G
S=ZZ/5[x1,x2,x3, MonomialOrder=>Lex]
sub(f1,S)%sub(I,S)
sub(f2,S)%sub(I,S)
sub(f3,S)%sub(I,S)
(
f1
,
G
)
(
f3
,
G
)
This time, we see that while
NF
and
NF
are the same as the normal
2
x
3
+
forms with the
grevlex
order, here we have that
NF
(
f2
,
G
)
=
x
3
−
2, where
is the Gröbner basis for
I
with respect to
lex
.
To see the explicit Gröbner basis that is being used, you can type
G
gens gb I
which returns the generators of the Gröbner basis as a matrix. To get the generators
in list form, type
flatten entries gens gb I
Exercise 3.9.
Based on your work in Exercise
3.7
, which variables do you expect
to be present in the normal forms? For example, do you expect
x
1
to appear in any
normal form?
Exercise 3.10.
Use Macaulay 2 to compute the normal forms of the polynomials
you found in Exercise
3.8
with respect to the ideals you found in Exercise
3.7
.Are
the resulting polynomials different when computed using
lex
,
grlex
,or
grevlex
?
Exercise 3.11.
Compute the set of standard monomials for the monomial orders
considered in Exercise
3.7
. You can use the following command in Macaulay 2 to
assist you:
leadTerm I
(This is the same as
leadTerm (gb I)
since a Gröbner basis is required.) What is
the relationship between thesemonomials and the terms in the normal forms computed
in Exercise
3.10
?
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