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An elimination order enables a certain variable to be removed from a system of equations. In the case of ideals in particular , it can be interesting to be able to calculate the intersection with some of the variables.
definition Be . A monomial order on the polynomial ring is fine for elimination if the following applies: . The leading term refers to the monomial order .
T
=
{
t
1
,
...
,
t
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}
⊂
{
x
1
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n
}
{\ displaystyle T = \ {t_ {1}, \ dotsc, t_ {s} \} \ subset \ {x_ {1}, \ dotsc, x_ {n} \}}
K
[
x
1
,
...
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n
]
{\ displaystyle K [x_ {1}, \ dotsc, x_ {n}]}
T
{\ displaystyle T}
L.
T
(
f
)
∈
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[
t
1
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,
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⇒
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∈
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[
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{\ displaystyle LT (f) \ in K [t_ {1}, \ dotsc, t_ {s}] \ Rightarrow f \ in K [t_ {1}, \ dotsc, t_ {s}]}
L.
T
{\ displaystyle LT}
Examples
The lexicographical order is an elimination order for all subsets .
{
x
k
,
...
,
x
n
}
,
k
∈
{
2
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,
n
}
{\ displaystyle \ {x_ {k}, \ dotsc, x_ {n} \}, k \ in \ {{2, \ dotsc, n} \}}
Block orders can also be used well as elimination orders .
Elimination rate Let an order of elimination for , an ideal, and a Gröbner basis of . Then: is Gröbner basis of .
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T
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{
t
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⊂
{
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{\ displaystyle T = \ {t_ {1}, \ dotsc, t_ {s} \} \ subset \ {x_ {1}, \ dotsc, x_ {n} \}}
I.
⊂
K
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{\ displaystyle I \ subset K [x_ {1}, \ dotsc, x_ {n}]}
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{\ displaystyle J = I \ cap K [t_ {1}, \ dotsc, t_ {s}]}
G
{\ displaystyle G}
I.
{\ displaystyle I}
F.
=
G
∩
K
[
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{\ displaystyle F = G \ cap K [t_ {1}, \ dotsc, t_ {s}]}
J
{\ displaystyle J}
Individual evidence
^ Sophia Feil: Gröbner bases and regularity. Retrieved July 31, 2019 .
^ Sophia Feil: Gröbner bases and regularity. Retrieved July 31, 2019 .
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ displaystyle K [x_ {1}, \ dotsc, x_ {n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46e8db7267506355a2bc7c036fc4617f3547a5ee)
![{\ displaystyle LT (f) \ in K [t_ {1}, \ dotsc, t_ {s}] \ Rightarrow f \ in K [t_ {1}, \ dotsc, t_ {s}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9bdec3f2d0fce84df5415416ecb7b3e69bf5a80)
![{\ displaystyle I \ subset K [x_ {1}, \ dotsc, x_ {n}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13d7f144ddff387e1fb1b4d5c472a56636c64377)
![{\ displaystyle J = I \ cap K [t_ {1}, \ dotsc, t_ {s}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51db80dcae649337060eeb5e839135934cb49c18)
![{\ displaystyle F = G \ cap K [t_ {1}, \ dotsc, t_ {s}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bb6bf316a29dc3da34dbcd4ecb5bb415cb6c153)
