implies that for all .
for all .
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 .
|Date of creation||2013-03-22 13:03:47|
|Last modified on||2013-03-22 13:03:47|
|Last modified by||mathcam (2727)|