
Gröbner basis

(Definition)


Definition of monomial orderings and support:
Let be a field, and let be the set of monomials in
, the polynomial ring in indeterminates. A monomial ordering is a total ordering on which satisfies
 implies that
for all
.
 for all .
In practice, for any
, we associate to the string
and compare monomials by looking at orderings on these tuples.
Example. An extremely prevalent example of a monomial ordering is given by the standard lexicographical ordering of strings. Other examples include graded lexicographic ordering and graded reverse lexicographic ordering.
Henceforth, assume that we have fixed a monomial ordering. Define the support of , denoted
, to be the set of terms with . Then define
.
A partial order on
:
We can extend our monomial ordering to a partial ordering on
as follows: Let
. If
, we say that if
.
It can be shown that:
 The relation defined above is indeed a partial order on
 Every descending chain
with
is finite.
A division algorithm for
:
We can then formulate a division algorithm for
:
Let
be an ordered tuple of polynomials, with
. Then for each
, there exist
with unique, such that

 For each
, does not divide any monomial in
.
Furthermore, if
for some , then
.
Definition of Gröbner basis:
Let be a nonzero ideal of
. A finite set
of polynomials is a Gröbner basis for if for all with there exists such that
.
Existence of Gröbner bases:
Every ideal
other than the zero ideal has a Gröbner basis. Additionally, any Gröbner basis for is also a basis of .

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Also defines: 
monomial ordering, Gröbner basis 
Crossreferences: basis, zero ideal, bases, ideal, divide, polynomials, division algorithm, finite, chain, relation, terms, fixed, orderings, string, associate, implies, satisfies, total ordering, indeterminates, polynomial ring, monomials, field, support
There are 4 references to this entry.
This is version 25 of Gröbner basis, born on 20020923, modified 20061004.
Object id is 3470, canonical name is GrobnerBasis.
Accessed 4526 times total.
Classification:
AMS MSC:  13P10 (Commutative rings and algebras :: Computational aspects of commutative algebra :: Polynomial ideals, Gröbner bases) 



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