Information Technology Reference
In-Depth Information
1.
f
i
[
f
j
]+
f
j
[
f
i
]=0
,
2.
f
i
[
f
i
]+[
f
i
]=0
.
For
D
= 5 they identify additionally two types of dependencies:
3.
f
i
[
f
j
]
x
k
+[
f
i
]
f
j
x
k
=0
4.
f
i
[
f
i
]
x
k
+[
f
i
]
x
k
=0
The authors conclude that these are the only existing dependencies for
D
=4and
D
= 5, and verify that their estimations are coherent with computer simulations.
For
D
=4thereare
2
ways of constructing relations on the form 1 and
m
ways
of constructing relations of the form 2. For
D
= 5 these numbers are multiplied
with the
n
+1 monomials of degree
≤
1 to form additional types of dependencies:
−
2
−
-
Case
D
=4:
I
=
R
m
(
n
+1)
2
−
-
Case
D
=5:
I
=
R
−
(
n
+1)
m
.
For
D
= 6 they state that the number of linearly independent equations is
n
2
+
n
1
+1
m
2
+
m
.
I
=
R −
·
(1)
At this point we step in and show where their analysis becomes wrong. They
conclude that the only relations will be multiples of
f
i
[
f
j
]+
f
j
[
f
i
] = 0. It seems
reasonable to assume that they have drawn this conclusion based on Buchberger's
two criterion, but this is not correct and will turn out fatal in further analysis
for larger
D
. Their formulas are indeed correct for 3
5, but for
D
=6
they forget to count the
dependencies among the dependencies
. By a slight abuse
of notation, which will be clarified in the next section, these dependencies may
be expressed as follows (the number of such dependencies is indicated in the
brackets to the right):
5.
f
i
[
f
j
[
f
k
]] + [
f
i
[
f
j
]]
f
k
+[[
f
i
]
f
k
]
f
j
=0(
3
)
6. [[
f
i
]
f
j
]
f
j
+[[
f
i
]
f
j
]=0(2
≤
D
≤
·
2
)
7. [[
f
i
]
f
i
]
f
i
+[[
f
i
]] = 0 (
1
)
This means we count
m
+
3
of the dependent equations twice, so we need to
balance this by calculating from an inclusion/exclusion point of view. Using the
authors notation, the correct bound should have been:
n
2
+
n
1
+1
m
2
+
m
+
m
3
+2
m
2
+
m
1
I
=
R
−
·
(2)
Note that the formulas above for 3
6 works only for quadratic equations.
The dependencies behave with respect to the degree
D
e
of the initial system
E
. If for instance
D
e
= 7 there would be constructed no dependencies applying
XL with
D
m
≤
≤
D
≤
6. If we work with linear equations, XL will introduce new
dependencies for each time we increase
D
m
.
