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A.3.1 Saturation

For any two ideals $i, j$ in the basering $R$ let

\begin{displaymath} \hbox{sat}(i,j)=\{x\in R\;\vert\; \exists\;n\hbox{ s.t. } x\cdot(j^n)\subseteq i\} = \bigcup_{n=1}^\infty i:j^n\end{displaymath}


denote the saturation of $i$ with respect to $j$. This defines, geometrically, the closure of the complement of V( $j$) in V( $i$) (where V( $i$) denotes the variety defined by $i$).

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 $m$ to be the homogeneous maximal ideal (note that maxideal(n) denotes the n-th power of the maximal ideal). Then $V(i)$ has no singular point outside the origin if and only if $sat(j+(f),m)$ 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