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A.4 Saturation
Since in the example above, the ideal
has the same vdim
in the polynomial ring and in the localization at 0 (each 195),
is smooth outside 0.
Hence
contains some power of the maximal ideal
. We shall
check this in a different manner:
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(
)
(V(
) denotes the variety defined by
).
In our case,
must be the whole ring, hence
generated by 1.
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. (Note that maxideal(n) denotes the
n-th power of the maximal ideal).
LIB "elim.lib"; // loading library elim.lib
// you should get the information that elim.lib has been loaded
// together with some other libraries which are needed by it
option(noprot); // no protocol
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
→ // LIB [0] string standard.lib,elim.li..., 83 char(s)
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