A little about Gröbner bases
The definition of a Gröbner basis is surprisingly easy to understand:
Fix a monomial order
on
, and let
be an ideal. A Gröbner basis for
(with respect to
with the property that for every nonzero
, the leading term of
is divisible by the leading term of
for some
.
As I understand, Gröbner bases help you answer the question: given a polynomial ideal , and a polynomial 
, is 
. This is a simple question in a univariate polynomial ring (I think 
) because you know is generated by a single polynomial, and so 
iff is divisible by the generator.
Things get hairier in multivariate polynomial rings (which partially explains why we didn’t cover this stuff in any of my abstract algebra classes)— there is no way to uniquely define division. First, there is no unique ordering of monomials the way that you have the ordering: 
in a univariate field, so you have to introduce the concept of a monomial ordering to even think about division. Then since we’re going to divide by multiple polynomials, the order of division matters. Like if 
, our division algorithm will return 
and 
such that
,
where none of the 
divide .
What we’d like is to have the same thing going for us in multivariate rings that we have in a univariate one: 
iff 
. In general, this is not true, because may be in , but for the particular basis choosen, none of the may divide . Gröbner bases are precisely those bases that have this nice property. And the Buchberger algorithm is the way to construct a Gröbner basis for an ideal given one of its bases.
Why is this useful? I don’t know, yet… How does this mysterious Buchberger algorithm work? I don’t know, yet… in fact, I don’t know how to pronounce ‘Buchberger’.
Possibly relevant posts:
- Kronecker’s Method of Factoring Polynomials over
![\Q[x]](/cz/latexrender/pictures/301e2bb0b6e9646cb427f264378415c9.png "\Q[x]")
(2/19/2005) - Varieties of Ideals (6/2/2005)
- Math GRE problem (2/11/2005)