source: git/Singular/LIB/weierstr.lib @ c7c5ef

//GMG, last modified 28.10.2001
///////////////////////////////////////////////////////////////////////////////
version="$Id: weierstr.lib,v 1.2 2006-05-19 11:50:35 Singular Exp $";
category="Teaching";
info="
LIBRARY:  weierstr.lib   Procedures for the Weierstrass Theorems
AUTHOR:                  G.-M. Greuel, greuel@mathematik.uni-kl.de

PROCEDURES:
 weierstrDiv(g,f,d);   perform Weierstrass division of g by f up to gegree d
 weierstrPrep(f,d);    perform Weierstrass preparation of f up to gegree d
 lastvarGeneral(f);    make f general of finite order w.r.t. last variable 
 generalOrder(f);      compute integer b s.t. f is x_n-general of order b 
           (parameters in square brackets [] are optional)
";

LIB "mondromy.lib";
LIB "poly.lib";
///////////////////////////////////////////////////////////////////////////////

proc generalOrder (poly f)
"USAGE:   generalOrder(f); f=poly
RETURN:  integer b if f is general of order b w.r.t. the last variable, say T,
         resp. -1 if not 
         (i.e. f(0,...,0,T) is of order b, resp. f(0,...,0,T)==0)
NOTE:    the procedure works for any monomial ordering
EXAMPLE: example generalOrder; shows an example
"
{ int ii;
  int n = nvars(basering);
  for (ii=1; ii<n; ii++)
  {
    f = subst(f,var(ii),0);
  }   
  return(mindeg(f));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y),ds;
   poly f = x2-4xy+4y2-2xy2+4y3+y4;
   generalOrder(f);
}  
///////////////////////////////////////////////////////////////////////////////

proc weierstrDiv ( poly g, poly f, int d )
"USAGE:   weierstrDiv(g,f,d); g,f=poly, d=integer
ASSUME:  f must be general of finite order, say b, in the last ring variable,
         say T; if not use the procedure lastvarGeneral first
PURPOSE: perform the Weierstrass division of g by f up to order d
RETURN:  a list, say l, of two polynomials and an interer, such that 
                   g = l[1]*f + l[2],  deg_T(l[2]) < b
         up to (including) total degree d
         l[3] is the number of iterations used
         if f is not T-general, return (0,g)
NOTE:    the procedure works for any monomial ordering
THEORY:  the proof of Grauert-Remmert (Analytische Stellenalgebren) is used 
         for the algorithm
EXAMPLE: example weierstrDiv; shows an example
"
{ 
//------------- initialisation and check T - general -------------------------
  int a,b,ii,D;
  poly r,h;
  list result;
  int y = printlevel - voice + 2;
  int n = nvars(basering);
  intvec v;
  v[n]=1;
  b = generalOrder(f);
  if (y>0) 
  {
     "//",f;"// is "+string(var(n))+"-general of order", b;
     pause("press <return> to continue");
  }
  if ( b==-1 )
  {
     "// second poly is not general w.r.t. last variable";
     "// use the procedure lastvarGeneral first";
     result=h,g;
     return(result);
  }
//------------------------- start computation --------------------------------
  D = d+b;
  poly fhat = jet(f,b-1,v);
  poly ftilde = (f-fhat)/var(n)^b;
  poly u = invunit(ftilde,D);
  if (y>0) 
  {
     "// fhat (up to order", d,"):";
     "//", fhat;
     "// ftilde:";
     "//", ftilde;
     "// ftilde-inverse:";
     "//", u;
     pause("press <return> to continue");
  }
  poly khat, ktilde;
  poly k=g;
  khat = jet(k,b-1,v);
  ktilde = (k-r)/var(n)^b;
  r = khat;
  h = ktilde;
  ii=0;
  while (size(k) > 0)
  {
  if (y>0) 
    {  
     "// loop",ii+1;
     "// khat:";
     "//", khat;
     "// ktilde:";
     "//", ktilde;
     "// remainder:";
     "//", r;
     "// multiplier:";
     "//", h;
     pause("press <return> to continue");
    }
    k = jet(-fhat*u*ktilde,D);
    khat = jet(k,b-1,v);
    ktilde = (k-khat)/var(n)^b;
    r = r + khat;
    h = h + ktilde;
    ii=ii+1;
  }  
  result = jet(u*h,d),jet(r,d),ii;
  return(result);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y),ds;
   poly f = y - xy2 + x2;
   poly g = y;
   list l = weierstrDiv(g,f,10); l;"";
   l[1]*f + l[2];               //g = l[1]*f+l[2] up to degree 10
}  
///////////////////////////////////////////////////////////////////////////////

proc weierstrPrep (poly f, int d)
"USAGE:   weierstrPrep(f,d); f=poly, d=integer
ASSUME:  f must be general of finite order, say b, in the last ring variable,
         say T; if not apply the procedure lastvarGeneral first
PURPOSE: perform the Weierstrass preparation of f up to order d
RETURN:  a list, say l, of two polynomials and one integer, 
         l[1] a unit, l[2] a Weierstrass polynomial, l[3] an integer 
         such that l[1]*f = l[2], where l[2] is a Weierstrass polynomial, 
         (i.e. l[2] = T^b + lower terms in T) up to (including) total degree d
         l[3] is the number of iterations used
         if f is not T-general, return (0,0)
NOTE:    the procedure works for any monomial ordering
THEORY:  the proof of Grauert-Remmert (Analytische Stellenalgebren) is used
         for the algorithm
EXAMPLE: example weierstrPrep; shows an example
"
{
  int n = nvars(basering);
  int b = generalOrder(f);
  if ( b==-1 )
  {
     "// second poly is not general w.r.t. last variable";
     "// use the procedure lastvarGeneral first";
     poly h,g;
     list result=h,g;
     return(result);
  }
  list L = weierstrDiv(var(n)^b,f,d);
  list result = L[1], var(n)^b - L[2],L[3];
  return(result);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,(x,y),ds;
   poly f = xy+y2+y4;
   list l = weierstrPrep(f,5); l; "";
   f*l[1]-l[2];                      // = 0 up to degree 5
}
///////////////////////////////////////////////////////////////////////////////

proc lastvarGeneral (poly f)
"USAGE:   lastvarGeneral(f,d); f=poly
RETURN:  poly, say g, obtained from f by a generic change of variables, s.t. 
         g is general of finite order b w.r.t. the last ring variable, say T
         (i.e. g(0,...,0,T)= c*T^b + higher terms, c!=0)
NOTE:    the procedure works for any monomial ordering
EXAMPLE: example lastvarGeneral; shows an example
"
{
  int n = nvars(basering);
  int b = generalOrder(f);
  if ( b >=0 )  { return(f); }
  else
  { 
    def B = basering;
    int ii;
    map phi;
    ideal m=maxideal(1);
    int d = mindeg1(f);
    poly g = jet(f,d);
    for (ii=1; ii<=n-1; ii++)
    {
       if (size(g)>size(subst(g,var(ii),0)) )
       { 
          m[ii]= var(ii)+ random(1-(voice-2)*10,1+(voice-2)*10)*var(n);
          phi = B,m;
          g = phi(f);
          break;
       }
    }
    if ( voice <=5 ) 
    { 
       return(lastvarGeneral(g));
    }
    if ( voice ==6 ) 
    { 
       for (ii=1; ii<=n-1; ii++)
      { 
         m[ii]= var(ii)+ var(n)*random(1,1000);
      }
      phi = basering,m;
      g = phi(f);
      return(lastvarGeneral(g));
    }
    else
    {
      for (ii=1; ii<=n-1; ii++)
      {
         m[ii]= var(ii)+ var(n)^random(2,voice*d);
      }
      phi = basering,m;
      g = phi(f);
      return(lastvarGeneral(g));
    }   
  }
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 2,(x,y,z),ls;
   poly f = xyz;
   lastvarGeneral(f);
}   
///////////////////////////////////////////////////////////////////////////////