Temporary Working Definition: A Gröbner basis of a set of multivariate polynomials
where 
(note 
necessarily), is an
equivalent but `more convenient' set of multivariate polynomials
where t may not be the same as n or m, necessarily. We usually find Gröbner bases by use of computer algebra.
More details will be given on what we mean by `equivalent' and `more convenient', later.
Remark. Gröbner bases are computed with respect
to a monomial ordering.
That is, we need to be able to compare
to xy or 
and decide which
is `first'.
There are several standard monomial orderings in the theory of Gröbner bases.
The two maple uses are
plex or `Pure Lexicographic' ordering. The
standard name for this is
simply `lexicographic' order.
It is familiar to everyone who can use a dictionary or
phone book.
Suppose we use the variables 
with the implicit
ordering 
.
This means 
is the `most important'
variable, say, down to 
, the `least important'.
(These emotionally-charged
words don't really mean anything, it's just a way of distinguishing).
Of course if
we start with x, y, z we can always rename so this
ordering reflects our priorities
(though often the choice of ordering among variables is arbitrary).
Now consider the monomial term 
, which we use as shorthand for
, and the monomial term
(which is a similar shorthand, of course).
We say that 
in the lexicographic order if the LEFTMOST nonzero entry
in the vector 
is POSITIVE.
Thus 
since 
.
So a lexicographic ordering counts powers of x
as being more important than powers
of other terms (making the identification
, of course).
tdeg or `total degree'.
The standard name for this is `graded reverse
lexicographic order', and the emphasis is on the first two words.
The `graded' means
that we consider total degree first:
if 
---that is, if the sum of the powers is larger,
then that's
the larger term.
If we have a tie in powers (e.g. 
vs 
) then we have
to resort to a tie-breaking rule:
it turns out to be useful to consider REVERSE
lexicographic order, which says that 
if the RIGHTMOST
nonzero entry of 
is NEGATIVE.
Note the double-negative from
pure lexicographic order, but it is the grading from the
sums of powers that turns
out to be most significant.
Then 
since the total degree
of the second term is higher.
Also, 
since 
.