Top
Back: Critical points
Forward: Long coefficients
FastBack: Examples
FastForward: Polynomial data
Up: Examples
Top: Singular 2-0-4 Manual
Contents: Table of Contents
Index: Index
About: About this document

A.4 Saturation

Since in the example above, the ideal $j+(f)$ has the same vdim in the polynomial ring and in the localization at 0 (each 195),

$f=0$ is smooth outside 0. Hence $j+(f)$ contains some power of the maximal ideal $m$. We shall check this in a different manner: 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( $i$) (V( $i$) denotes the variety defined by $i$). In our case, $sat(j+(f),m)$ must be the whole ring, hence generated by 1.

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. (Note that maxideal(n) denotes the n-th power of the maximal ideal).

 
  LIB "elim.lib";         // loading library elim.lib
  // you should get the information that elim.lib has been loaded
  // together with some other libraries which are needed by it
  option(noprot);         // no protocol
  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
==> // LIB                  [0]  string standard.lib,elim.li..., 83 char(s)


Top Back: Critical points Forward: Long coefficients FastBack: Examples FastForward: Polynomial data Up: Examples Top: Singular 2-0-4 Manual Contents: Table of Contents Index: Index About: About this document
            User manual for Singular version 2-0-4, May 2003, generated by texi2html.