Top
Back: A.3 Critical points
Forward: A.5 Long coefficients
FastBack: Appendix A Examples
FastForward: Appendix B Polynomial data
Up: Appendix A Examples
Top: 1 Preface
Contents: Table of Contents
Index: F 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

                           n        ∞⋃     n
sat(i,j) = {x ∈ R | ∃ n s.t. x ⋅(j ) ⊆ i} = i : j
                                    n=1

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: A.3 Critical points Forward: A.5 Long coefficients FastBack: Appendix A Examples FastForward: Appendix B Polynomial data Up: Appendix A Examples Top: 1 Preface Contents: Table of Contents Index: F Index About: About This Document
            User manual for Singular version 2-0-4, October 2002, generated by texi2html.