4.4 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
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)