Biology Reference
In-Depth Information
To use Groebner bases to diagonalize the system, we first need to rewrite the equations
to ensure that the right-hand side is zero:
x
2
y
2
z
2
+
+
−
1
=
0
x
2
z
2
+
−
y
=
0
(1.12)
x
−
z
=
0
.
Next, we need to find the Groebner basis for the functions that form the left-hand
sides of the equations:
h
1
x
2
y
2
z
2
x
2
z
2
z
.
To compute the Groebner basis for these functions in
SAGE
, we use the following
commands (click on “evaluate” after entering each line)
2
:
=
+
+
−
1;
h
2
=
+
−
y
;
h
3
=
x
−
P.<x,y,z> = PolynomialRing(RR, 3, order='lex')
I = ideal(x ˆ 2+y ˆ 2+z ˆ 2-1, x ˆ 2+z ˆ 2-y, x-z)
B = I.groebner_basis()
In the first command, the variables in the system are listed on the left. On the right, RR
denotes the real numbers (meaning we want to work over the reals) and three is the
number of variables. For our purposes, we will always need to use
order='lex'
.
After evaluating the last command,
SAGE
returns the Groebner basis:
[x-z, y - 2*zˆ2, zˆ4 + 1/2*zˆ2 - 1/4].
It follows from the general theory outlined in the online Appendix 1 that the
systems of Eqs. (
1.12
) has a solution set equivalent to the solution set of the system
1
2
z
2
1
4
=
z
4
+
−
0
2
z
2
y
−
=
0
x
−
z
=
0
.
Notice that this system is “diagonal” in the sense that if we solve the first equation for
z
,
we can use the values in the second equation to solve for
y
and then
in the third equation
for
x
. After doing this, we obtain the solutions
z
−
1
+
√
5
4
2
z
2
=±
;
y
=
;
x
=
z
for
−
1
+
√
5
4
−
1
+
√
5
4
+
√
5
2
−
1
the system,
which le
ads to the solutions
z
=
;
y
=
;
x
=
−
1
+
√
5
4
−
1
+
√
5
4
=
−
1
+
√
5
2
and
z
=−
;
y
;
x
=−
.
2
Note that exponentiation is indicated with ˆ and multiplication must be indicated with
∗
.
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