There are several ways to look at the theory of Gröbner basis. One way is the following: Suppose we
are given polynomials in one variable over say
and we would like to know whether
another polynomial
belongs to the ideal generated by the
’s. A good way to decide the issue would
be to first compute the gcd
of
,
, …,
and check whether
is a multiple of
. One can
achieve this by doing the long division of
by
and checking whether the remainder is zero.
All this is possible because
is a Euclidean domain and also a principle ideal domain
(PID) wherein any ideal is generated by a single element. Therefore we have essentially just one
polynomial - the gcd - which generates the ideal generated by
. The ring of
integers or the ring of polynomials in one variable over any field are examples of PIDs whose
ideals are generated by single elements. However, when we consider more general rings (not
PIDs) like the one we are dealing with here, we do not have a single gcd but a set of several
polynomials which generates an ideal in general. A Gröbner basis of an ideal can be thought
of as a generalization of the gcd. In the univariate case, the Gröbner basis reduces to the
gcd.
Gröbner basis theory generalizes these ideas to multivariate polynomials which are neither Euclidean
rings nor PIDs. Since there is in general not a single generator for an ideal, Gröbner basis theory comes up
with the idea of dividing a polynomial with a set of polynomials, the set of generators of the ideal, so that
by successive divisions by the polynomials in this generating set of the given polynomial, the remainder
becomes zero. Clearly, every generating set of polynomials need not possess this property. Those special
generating sets that do possess this property (and they exist!) are called Gröbner bases. In order for a
division to be carried out in a sensible manner, an order must be put on the ring of polynomials, so that the
final remainder after every division is strictly smaller than each of the divisors in the generating set. A
natural order exists on the ring of integers or on the polynomial ring ; the degree of the
polynomial decides the order in
. However, even for polynomials in two variables there is no
natural order a priori (is
greater or smaller than
?). But one can, by hand as
it were, put an order on such a ring by saying
, where
is an order, called the
lexicographical order. We follow this type of order,
and ordering polynomials
by considering their highest degree terms. It is possible to put different orderings on a given
ring which then produce different Gröbner bases. Clearly, a Gröbner basis must have ‘small’
elements so that division is possible and every element of the ideal when divided by the Gröbner
basis elements leaves zero remainder, i.e. every element modulo the Gröbner basis reduces to
zero.
In the literature, there exists a well-known algorithm called the Buchberger algorithm which may be
used to obtain the Gröbner basis for a given set of polynomials in the ring. So a Gröbner basis of can
be obtained from the generators
given in Equation (28
) using this algorithm. It is essentially again a
generalization of the usual long division that we perform on univariate polynomials. More conveniently, we
prefer to use the well known application Mathematica. Mathematica yields a 3-element Gröbner basis
for
:
This Gröbner basis of the ideal is then used to obtain the generators for the module of syzygies.
Note that although the Gröbner basis depends on the order we choose among the
, the module itself
is independent of the order [2
].
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