Top
Back: C.3 Syzygies and resolutions
Forward: C.5 Gauss-Manin connection
FastBack: Appendix C Mathematical background
FastForward: Appendix D SINGULAR libraries
Up: Appendix C Mathematical background
Top: 1 Preface
Contents: Table of Contents
Index: F Index
About: About This Document

C.4 Characteristic sets

Let < be the lexicographical ordering on R = K[x1,...,xn] with x1 < ... < xn. For f R let lvar(f) (the leading variable of f) be the largest variable in f, i.e., if f = as(x1,...,xk1)xks + ... + a0(x1,...,xk1) for some k n then lvar(f) = xk.

Moreover, let ini(f) := as(x1,...,xk1). The pseudo remainder r = prem(g,f) of g with respect to f is defined by the equality ini(f)a g = qf + r with deglvar(f)(r) < deglvar(f)(f) and a minimal.

A set T = {f1,...,fr}⊂ R is called triangular if lvar(f1) < ... < lvar(fr). Moreover, let U T, then (T,U) is called a triangular system, if T is a triangular set such that ini(T) does not vanish on V (T) \ V (U)(=: V (T \ U)).

T is called irreducible if for every i there are no di,fi‘,fi“ such that

lvar(di) < lvar(fi) = lvar(fi‘) = lvar(fi“),
0 ⁄∈ prem ({di,ini(fi‘),ini(fi“)},{f1,...,fi−1}),
prem(difi − fi‘fi“,{f1,...,fi− 1}) = 0.
Furthermore, (T,U) is called irreducible if T is irreducible.

The main result on triangular sets is the following: let G = {g1,...,gs}⊂ R then there are irreducible triangular sets T1,...,Tl such that V (G) = i=1l(V (Ti \ Ii)) where Ii = {ini(f)f Ti}. Such a set {T1,...,Tl} is called an irreducible characteristic series of the ideal (G).

Example:
  ring R= 0,(x,y,z,u),dp;
  ideal i=-3zu+y2-2x+2,
          -3x2u-4yz-6xz+2y2+3xy,
          -3z2u-xu+y2z+y;
  print(char_series(i));
→ _[1,1],3x2z-y2+2yz,3x2u-3xy-2y2+2yu,
→ x,     -y+2z,      -2y2+3yu-4       

Top Back: C.3 Syzygies and resolutions Forward: C.5 Gauss-Manin connection FastBack: Appendix C Mathematical background FastForward: Appendix D SINGULAR libraries Up: Appendix C Mathematical background 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 texinfo.