Definition 1. Fix an integer
1<= k<= n. We say that a monomial order
> on K[x1, x2,...,
xn] is of k-elimination
type provided that any monomial involving one of x1,
x2,..., xk
is greater than all monomials in K[xk+1,
xk+2,..., xn]. |
This is the kth
elimination order of Bayer and Stillman and it is
now supported by the Groebner
applet.
It is preferred to lex order when eliminating only
certain variables from a system of equations since it
leads to simpler bases.Examples.
- (2,0,1) >1
(1,2,0) since the first entry in (2,0,1) is greater than the first entry in (1,2,0). Hence
x2z >1
xy2.
(1,2,0) >2
(2,0,1) since the sum of the first two entries in (1,2,0) is greater than the
sum of the first two entries in (2,0,1). Hence
xy2 >2
x2z.
(1,2,0) >3
(2,0,1) since the sum of the first three entries in (1,2,0)
equals the
sum of the first three entries in (2,0,1) and (1,2,0) >grevlex
(2,0,1). Hence
xy2 >3
x2z.
- Consider the equations
f1
= t2 + x2
+ y2 + z2
= 0
f2
= t2 + 2x2
- xy - z2 = 0
f3
= t + y3 -
z3 = 0.
We want to eliminate t from these
equations.
- The reduced Groebner
basis for I = <f1,
f2,
f3>
w.r.t. lex order with t>x>y>z
is given by
g1
= y12
- 4z3y9
+ 5y8 +
6z6y6
+ 6z2y6
- 10z3y5
+ 5y4 -
4z9y3
- 12z5y3
+ 5z6y2
+ 13z2y2
+ z12 +
6z8 + 9z4
g2
= xz6
+ 3xz2 -
y11 + 4y8z3
- 5y7 -
5y5z6
- 3y5z2
+ 10y4z3
- 5y3 +
2z9y2
+ 6y2z5
- 3yz6 -
7yz2
g3
= xy + y6
- 2y3z3
+ 2y2 +
z6 + 3z2
g4
= x2
+ y6 -
2y3z3
+ y2 + z6
+ z2
g5
= t + y3
- z3.
Hence a (Groebner) basis for the 1st
elimination ideal I I K[x,
y, z] is <g1,
g2,
g3,
g4>:
we have four generators, one with total degree 12.
- The reduced Groebner
basis for I w.r.t. the 1st
elimination order with t>x>y>z
is
h1
= x2
- xy - y2
- 2z2
h2
= y6
- 2y3z3
+ z6 + xy
+ 2y2 +
3z2
h3
= t + y3
- z3.
We have got a much simpler basis
for I I
K[x, y, z]:
<h1,
h2>.
This is the reduced Groebner basis of the
1st elimination ideal I I K[x,
y, z] w.r.t.
grevlex order.
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