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
sat(i,j) = {x in R | there is an n s.t. x*(j^n) contained in i}
= union_(n=1...) of i:j^n,
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..., 52 char( ==> s)