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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.

 


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