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3.3. NONLINEAR MULTIVARIATE EQUATIONS: DISTRIBUTED 79
x k+1 , . . . , x n in a Gröbner base computed with such an order are all that
can be deduced about these variables. It is common, but by no means
required, to use tdeg for both > ′ and > ′′ . Note that this is not the
same as simply using tdeg, since the exponents of x k+1 , . . . , x n are not
considered unless x 1 , . . . , x k gives a tie.
weighted orderings Here we compute the total degree with a weighting factor,
e.g. we may weight x twice as much as y, so that the total degree of
x i y j would be 2i + j. This can come in lexicographic or reverse lexicographic
variants.
matrix orderings These are in fact the most general form of orderings [Rob85].
Let M be a fixed n × n matrix of reals, and regard the exponents of A as
an n-vector a. Then we compare A and B by computing the two vectors
M.a and M.b, and comparing these lexicographically.
lexicographic M is the identity matrix.
⎛
⎞
1 1 . . . 1 1
1 0 . . . 0 0
. grlex M =
0 .. 0 . . . 0
.
⎜
⎟
⎝ . . . . . ⎠
0 . . . 0 1 0
tdeg It
⎛
would be tempting to
⎞
say, by analogy with grlex, that the matrix
1 1 . . . 1 1
0 0 . . . 0 1
is
0 . . . 0 1 0
⎜
⎟
⎝
. . . . .
⎠ . However, this is actually grlex with the
0 1 0 . . . 0
variable order reversed, not ⎛genuine reverse lexicographic. ⎞ To get
1 1 . . . 1 1
−1 0 . . . 0 0
. that, we need the matrix
0 .. 0 . . . 0
⎜
⎟
, or, if we are
⎝ . . . . . ⎠
0 . . . 0 −1 0
adopting
⎛
the Maple convention
⎞
of reversing the variables as well,
1 1 . . . 1 1
0 0 . . . 0 −1
0 . . . 0 −1 0
⎜
⎟
⎝
. . . . .
⎠ .
0 −1 0 . . . 0
k-elimination ( If the matrices ) are M k for > ′ and M n−k for > ′′ , then
Mk 0
M =
.
0 M n−k
weighted orderings Here the first row of M corresponds to the weights,
instead of being uniformly 1.