Graphics Reference
In-Depth Information
Another application using Gröbner bases is:
A test for ideal membership:
To see if a polynomial f belongs to an ideal, compute a
Gröbner basis G for the ideal and check if the G-normal form for f is zero.
Here are some more uses of Gröbner bases:
10.10.17. Example.
To find the solutions
S
to the equations
2
2
2
xyz
xz y
xy
+-=
-=
=
1
2
2
2
.
Solution.
If
2
2
2
2
2
2
pxyz
=+--
1
,
pxyz dpxy
=--
,
=-
,
1
2
3
then by Lemma 10.8.2,
S
= V(p
1
,p
2
,p
3
) = V(I), where I =<p
1
,p
2
,p
3
>. Using our algo-
rithms one can show that G = {g
1
,g
2
,g
3
} is a reduced Gröbner basis for I, where
2
gxy
gyy
gz
=-
=+-
=
1
2
1
2
2
.
3
Since I =<g
1
,g
2
,g
3
>, the interesting thing about this is that g
2
and g
3
are polynomials
of only
one
variable. Solving for their zeros we get
z
=
0
-±
15
2
y
=
.
Substituting these solutions into the equation g
1
= 0 and solving for x gives
-15
2
x =±
.
These values give us all the answers over the complex numbers. Although our example
is a trivial one, we shall see that it is not atypical and that Gröbner bases can be used
to eliminate variables in equations, solve the simpler equations, and then extend the
partial solutions to a solution of the original equations.
10.10.18. Example.
To find an implicit form for the curve parameterized by
4
xt
yt
zt
=
=
=
3
2
.
