# Singular

### 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