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We find that
f
1
x
1
,
x
3
=
4
r
1
+
3
r
2
+
0
r
3
=
x
1
+
x
2
,
x
1
+
3
.
So the set
F
1
of polynomials that fit the data for node
x
1
is
F
1
=
f
1
+
I
.Usingthe
above strategy, we can find similar sets
F
2
=
f
2
+
I
and
F
3
=
f
3
+
I
for
x
2
and
x
3
,
3
x
1
+
2
x
1
+
respectively, where
f
2
(
4.
The data set in the previous example is called a
time series
, since the data points
are considered to be successive. We call
F
1
×···×
x
)
=
x
1
+
4 and
f
3
(
x
)
=
2
x
1
+
F
n
the
model space
associated to
the given data and any PDS
F
n
is called a
model
. Note that
the ideal
I
of vanishing polynomials is independent of the node or particular function.
For the following exercises, consider the time series over
(
f
1
,...,
f
n
)
∈
F
1
×···×
Z
3
, which contains five
observations for a 3-node system.
111
201
200
022
022
Exercise 3.7.
Compute the ideal of the data points using Macaulay 2 and the
lex
monomial order with the variable order
x
1
x
3
. Use the previous example as
a reference. In Macaulay 2, polynomial rings require an explicit choice of monomial
ordering. For this exercise, you will use the following:
x
2
R=ZZ/3[x1,x2,x3,MonomialOrder=>Lex]
If none is given, then
grevlex
is used by default. Also the order of the variables in the
polynomial ring indicates the variable order. Is the ideal different under the
grevlex
order?
n
be a data set as in Example
3.6
.A
Exercise 3.8.
Let
D
={
c
1
,...,
c
m
}⊆F
n
polynomial
f
: F
→ F
that interpolates
D
can be constructed as
m
n
j
=
1
(
p
−
1
f
(
x
1
,...,
x
n
)
=
f
(
c
i
1
,...,
c
in
)
1
−
(
x
j
−
c
ij
)
).
i
=
1
Compute an interpolating polynomial for each node using this interpolation formula
and the one provided in the text above. Do they differ on the data? Do they differ on
other data points?
Project 3.4.
An easy calculation shows that there are
p
n
states in
n
.Fromthis
(
Z
p
)
we find that there are
p
p
n
polynomials in
Z
p
[
x
1
,...,
x
n
]
.Howmany
n
-dimensional
n
, how many models
PDSs are there over
Z
p
[
x
1
,...,
x
n
]
? Given a data set
D
⊂
(
Z
p
)
are there in the corresponding model space?
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