Biology Reference
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FIGURE 3.2
The Gröbner fan of the ideal in Example
3.10
intersected with the standard 2-simplex.
The
x
1
-axis is on the right, the
x
2
-axis on the left, and the
x
3
-axis at the top.
cone volume when bounded by an
n
-sphere centered at the cone's vertex. While com-
puting the exact relative cone volumes requires knowing the facets of the fan, that is,
the fan itself, approximation of the relative volumes in many cases is sufficient [
33
]. A
stochastic method for estimating the relative volumes of the Gröbner cones of a Gröb-
ner fan without computing the actual fan, as well as a Macaulay 2 implementation
for uniform sampling from the Gröbner fan, is presented in [
34
].
Exercise 3.15.
In Example
3.10
, theweight vector
ω
1
={
2
,
1
,
1
}
generatedGröbner
z
2
y
2
x
2
basis
G
1
={
−
z
,
−
y
,
xz
+
yz
−
x
−
y
−
z
+
1
,
xy
−
yz
,
−
x
}
. Can you find
a different weight vector that produces the same Gröbner basis?
Exercise 3.16.
Which one of the three Gröbner bases in Example
3.10
corresponds
to the lexicographic monomial ordering?
Exercise 3.17.
be a Boolean
PDS that fits a data set. Let
I
be the ideal of functions that vanish on the data points.
Suppose 20% of all monomial orderings generated the following normal form of
F
with respect to
I
:
Let us construct a stochastic PDS. Let
F
=
(
f
1
,
f
2
)
f
1
=
x
1
+
x
2
,
x
1
,
and 80% of the monomial orderings generated
f
2
=
f
1
=
x
1
,
f
2
=
x
1
x
2
.
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