| ### A.3.1 Saturation For any two ideals 444#444 in the basering 53#53 let 445#445 denote the saturation of 57#57 with respect to 55#55. This defines, geometrically, the closure of the complement of V( 55#55) in V( 57#57) (where V( 57#57) denotes the variety defined by 57#57). 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. We apply saturation to show that a variety has no singular points outside the origin (see also Critical points). We choose 297#297 to be the homogeneous maximal ideal (note that `maxideal(n)` denotes the n-th power of the maximal ideal). Then 446#446 has no singular point outside the origin if and only if 447#447 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
      // list the variables defined so far:
      listvar();
    ==> // r2                             [0]  *ring
    ==> //      j                              [0]  ideal, 3 generator(s)
    ==> //      f                              [0]  poly

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