### 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 `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 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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