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A.3.1 Saturation
For any two ideals
in the basering
let
denote the saturation of
with respect to
. This defines,
geometrically, the closure of the complement of V(
) in V(
)
(where V(
) denotes the variety defined by
).
The saturation is computed by the procedure sat in
elim.lib by computing iterated ideal quotients with the maximal
ideal. sat returns a list of two elements: the saturated ideal
and the number of iterations.
We apply saturation to show that a variety has no singular points
outside the origin (see also Critical points).
We choose
to be the homogeneous maximal ideal
(note that maxideal(n) denotes the n-th power of the maximal
ideal).
Then
has no singular point outside the origin
if and only if
is the whole ring, that is, generated by 1.
LIB "elim.lib"; // loading library elim.lib
ring r2 = 32003,(x,y,z),dp;
poly f = x^11+y^5+z^(3*3)+x^(3+2)*y^(3-1)+x^(3-1)*y^(3-1)*z3+
x^(3-2)*y^3*(y^2)^2;
ideal j=jacob(f);
sat(j+f,maxideal(1));
==> [1]:
==> _[1]=1
==> [2]:
==> 17
// list the variables defined so far:
listvar();
==> // r2 [0] *ring
==> // j [0] ideal, 3 generator(s)
==> // f [0] poly
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