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A.4 Saturation
Since in the example above, the ideal
j + (f)
has the same vdim
in the polynomial ring and in the localization at 0 (each 195),
f = 0
is smooth outside 0.
Hence
j + (f)
contains some power of the maximal ideal
m
. We shall
check this in a different manner:
For any two ideals
i,j
in the basering
R
let
denote the saturation of
i
with respect to
j
. This defines,
geometrically, the closure of the complement of V(
j
) in V(
i
)
(V(
i
) denotes the variety defined by
i
).
In our case,
sat(j + (f),m)
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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