|  |  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 satinelim.libby computing iterated ideal quotients with the maximal
ideal.satreturns 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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