Usage:
<x_to_last x f g
If R is the base ring of x and S is the current ring, the
script returns inverse ring maps f: R -->S and g: S-->R. The
effect of f is to make the linear forms in "x" act as the
last variables. Thus after
<x_to_last x s f g
ev f m m1,
the last variable of S will act on m1 the way the linear
form which is the last entry of x acts on m.
Parameters:
x = 1 x m matrix of linear forms over a ring with no more
variables than the current ring.
Output values:
f = ideal of the current ring with m linear forms, where m
is the number of variables of the base ring of x
g = ideal of the base ring of x, again with m variables.
typical usage: To saturate the ideal i (defined over r,
with last variable z and reverse lex order)
with respect to a linear form x we might do
<x_to_last x f g
ev f i i1
std i1 i1
sat i1 i1 z
ev g i1 i1
Useful for checking whether x is a nonzerodivisor, using sat.
See NZD.
The idea is to find an nxn scalar invertible matrix G
whose last m columns consist of the coefficients of x in terms
of the variables of R. The map g is defined by the ideal
(z1, ... zn)G, where the zi are the variables of R.
If we set,
F := G^(-1)
then the map f is defined by the ideal
(y_1,...,y_n)F, where the y_i are the variables of S.