source: git/Singular/LIB/dmodapp.lib @ 04923be

//////////////////////////////////////////////////////////////////////////////
version="$Id: dmodapp.lib,v 1.1 2007-12-11 12:00:01 levandov Exp $";
category="Noncommutative";
info="
LIBRARY: dmodapp.lib     Applications of algebraic D-modules
AUTHORS: Viktor Levandovskyy,     levandov@math.rwth-aachen.de
@*            Seminar Group (Lehrstuhl B and D fuer Mathematik, RWTH Aachen)

THEORY: Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R,
@*      one is interested in the R[1/F]-module of rank one, generated by F^s
@*      for a natural number s.

GUIDE:
@* - Ann F^s = I = I(F^s) = LD in D(R)[s] can be computed by SannfsBM, SannfsOT, SannfsLOT
@* - global Bernstein polynomial bs resp. BS in K[s] can be computed by bernsteinBM

MAIN PROCEDURES:

charVariety(I);       compute the characteristic variety of the ideal I

SECONDARY PROCEDURES FOR D-MODULES:

foo();                    dummy prototype for a future procedure

AUXILIARY PROCEDURES:

Appell(a,b,c,d);      create an ideal annihilating Appel F4 function

SEE ALSO: dmod_lib, gmssing_lib
";

LIB "poly.lib";
LIB "sing.lib";
LIB "primdec.lib";
LIB "dmod.lib"; // loads e.g. nctools.lib

proc charVariety(ideal I)
"USAGE:  charVariety(I);  I an ideal
RETURN:  ring
PURPOSE: compute the D-module structure of basering[1/f]*f^s with the algorithm given in S and with the Groebner basis engine given in 'eng'
ASSUME: the ground ring is the Weyl algebra with x's before d's
NOTE:    activate the output ring with the @code{setring} command.
@*       In the output (in a commutative ring):
@*       - the ideal CV is the characteristic variety char(I)
@*       If @code{printlevel}=1, progress debug messages will be printed,
@*       if @code{printlevel}>=2, all the debug messages will be printed.
EXAMPLE: example annfs; shows examples
"
{
  // 1. introduce the weights 0, 1
  def save = basering;
  list LL = ringlist(save);
  list L;
  int i; 
  for(i=1;i<=4;i++)
  {
    L[i] = LL[i];
  }
  list OLD = L[3];
  list NEW; list tmp;
  tmp[1] = "a";  // string
  intvec iv;
  int N = nvars(basering); N = N div 2;
  for(i=N+1; i<=2*N; i++)
  {
    iv[i] = 1;
  }
  tmp[2] = iv;
  NEW[1] = tmp;
  for (i=2; i<=size(OLD);i++)
  {
    NEW[i] = OLD[i-1];
  }
  L[3] = NEW;
  def @U = ring(L);
  // 2. create the commutative ring  
  setring save;
  list CL;
  for(i=1;i<=4;i++)
  {
    CL[i] = L[i];
  }
  CL[3] = OLD;
  def @CU = ring(CL);
  // comm ring is ready
  setring @U;
  // 2. compute Groebner basis
  ideal I = imap(save,I);
  //  I = groebner(I);
  I = slimgb(I);
  setring @CU;
  ideal CV = imap(@U,I);
  //  CV = groebner(CV); // cosmetics
  CV = slimgb(CV);
  export CV;
  kill @U;
  return(@CU);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),Dp;
  poly F = x3-y2;
  printlevel = 0;
  def A  = annfs(F);
  setring A; // Weyl algebra
  LD; // the ideal
  def CA = charVariety(LD);
  setring CA;
  CV;
  dim(CV);
}

proc charInfo(ideal I)
"USAGE:  charInfo(I);  I an ideal 
RETURN:  ring
PURPOSE: compute the characteristic information for I
ASSUME: the ground ring is the Weyl algebra with x's before d's
NOTE:    activate the output ring with the @code{setring} command.
@*       In the output (in a commutative ring):
@*       - the ideal CV is the characteristic variety char(I)
@*       - the ideal SL is the singular locus of char(I)
@*       - the list PD is the primary decomposition of char(I)
@*       If @code{printlevel}=1, progress debug messages will be printed,
@*       if @code{printlevel}>=2, all the debug messages will be printed.
EXAMPLE: example annfs; shows examples
"
{
  def save = basering;
  def @A = charVariety(I);
  setring @A;
  // run slocus
  // run primdec
}


proc Appel(number a,b,c,d)
{
  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),dp);
  matrix @D[4][4];
  @D[1,3]=1; @D[2,4]=1;
  def @S = nc_algebra(1,@D);
  setring @S;
  ideal IAppel =
    Dx*(x*Dx+c-1) - x*(x*Dx+y*Dy+a)*(x*Dx+y*Dy+b),
    Dy*(y*Dy+d-1) - y*(x*Dx+y*Dy+a)*(x*Dx+y*Dy+b);
  export IAppel;
  kill @r;
  return(@S);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  def Ap = Appel(1,2,3,4);
  setring Ap;
  IAppel;
}

proc isFsat(ideal I, poly F)
{
  /* checks whether I is F-saturated, that is Ke  (D -F-> D/I) is 0 */
  /* works in any algebra */
  /*  for simplicity : later check attrib */
  /* returns -1 if true */
  I = groebner(I);
  matrix @M = matrix(I);
  matrix @F[1][1] = F;
  module S = modulo(@F,@M);
  S = NF(S,I);
  S = groebner(S);
  return( (gkdim(S) == -1) );
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly G = x*(x-y)*y;
  def A = annfs(G);
  setring A;
  poly F = x3-y2;
  isFsat(LD,F);
  ideal J = LD*F;
  isFsat(J,F);
}

proc DLoc(ideal I, poly F)
"USAGE:  DLoc(I, F);  I an ideal, F a poly
RETURN: nothing (exports objects instead)
ASSUME: the basering is a Weyl algebra
PURPOSE: compute the presentation of the localization of D/I w.r.t. f^s
NOTE:    In the basering, the following objects are exported:
@*       - the ideal LD0 (which is a Groebner basis) is the presentation of the localization
@*       - the ideal BS contains the roots with multiplicities of a Bernstein polynomial of D/I w.r.t f.
@*       If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example DLoc; shows examples
"
{
  /* runs SDLoc and DLoc0 */
  /* assume: run from Weyl algebra */
  int old_printlevel = printlevel;
  printlevel=printlevel+1;
  def @R = basering;
  def @R2 = SDLoc(I,F);
  setring @R2;
  poly F = imap(@R,F);
  def @R3 = DLoc0(LD,F);
  setring @R3;
  ideal bs = BS[1];
  intvec m = BS[2];
  setring @R;
  ideal LD0 = imap(@R3,LD0);
  export LD0;
  ideal bs = imap(@R3,bs);
  list BS; BS[1] = bs; BS[2] = m;
  export BS;
  kill @R3;
  printlevel = old_printlevel;
}
example;
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def R = Weyl();    setring R;
  poly F = x2-y3;
  ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F; 
  DLoc(I, x2-y3);
  LD0; 
  BS;
}


proc DLoc0(ideal I, poly F)
"USAGE:  DLoc0(I, F);  I an ideal, F a poly
RETURN:  ring
PURPOSE: compute the presentation of the localization of D/I w.r.t. f^s, where D is a Weyl Algebra, based on the output of procedure SDLoc
ASSUME: the basering is similar to the output ring of SDLoc procedure
NOTE:    activate this ring with the @code{setring} command. In this ring,
@*       - the ideal LD0 (which is a Groebner basis) is the presentation of the localization
@*       - the ideal BS contains the roots with multiplicities of a Bernstein polynomial of D/I w.r.t f.
@*       If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example DLoc0; shows examples
"
{
  /* assume: to be run in the output ring of SDLoc */
  /* todo: add F, eliminate vars*Dvars, factorize BS */
  /* analogue to annfs0 */
  def @R2 = basering;
  // we're in D_n[s], where the elim ord for s is set
  ideal J = NF(I,std(F));
  // make leadcoeffs positive
  int i;
  for (i=1; i<= ncols(J); i++)
  {
    if (leadcoef(J[i]) <0 )
    {
      J[i] = -J[i];
    }
  }
  J = J,F;
  ideal M = groebner(J);
  int Nnew = nvars(@R2);
  ideal K2 = nselect(M,1,Nnew-1);
  int ppl = printlevel-voice+2;
  dbprint(ppl,"// -1-1- _x,_Dx are eliminated in basering");
  dbprint(ppl-1, K2);
  // the ring @R3 and the search for minimal negative int s
  ring @R3 = 0,s,dp;
  dbprint(ppl,"// -2-1- the ring @R3 = K[s] is ready");
  ideal K3 = imap(@R2,K2);
  poly p = K3[1];
  dbprint(ppl,"// -2-2- attempt the factorization");
  list PP = factorize(p);          //with constants and multiplicities
  ideal bs; intvec m;             //the Bernstein polynomial is monic, so we are not interested in constants
  for (i=2; i<= size(PP[1]); i++)  //we delete P[1][1] and P[2][1]
  {
    bs[i-1] = PP[1][i];
    m[i-1]  = PP[2][i];
  }
  ideal bbs; int srat=0; int HasRatRoots = 0;
  int sP;
  for (i=1; i<= size(bs); i++) 
  {
    if (deg(bs[i]) == 1)
    {
      bbs = bbs,bs[i];
    }
  }
  if (size(bbs)==0)
  {
    dbprint(ppl-1,"// -2-3- factorization: no rational roots");
    //    HasRatRoots = 0;
    HasRatRoots = 1; // s0 = -1 then
    sP = -1;
    // todo: return ideal with no subst and a b-function unfactorized
  }
  else
  {
    // exist rational roots
    dbprint(ppl-1,"// -2-3- factorization: rational roots found");
    HasRatRoots = 1;
    //    dbprint(ppl-1,bbs);
    bbs = bbs[2..ncols(bbs)];
    ideal P = bbs;
    dbprint(ppl-1,P);
    srat = size(bs) - size(bbs);
    // define minIntRoot on linear factors or find out that it doesn't exist
    intvec vP;
    number nP;
    P = normalize(P); // now leadcoef = 1
    P = lead(P)-P;
    sP = size(P);
    int cnt = 0;
    for (i=1; i<=sP; i++)
    {
      nP = leadcoef(P[i]);
      if ( (nP - int(nP)) == 0 )
      {
        cnt++;
        vP[cnt] = int(nP);
      }
    }
//     if ( size(vP)>=2 )
//     {
//       vP = vP[2..size(vP)];
//     }
    if ( size(vP)==0 )
    {
      // no roots!
      dbprint(ppl,"// -2-4- no integer root, setting s0 = -1");
      sP = -1; 
      //      HasRatRoots = 0; // older stuff, here we do substitution
      HasRatRoots = 1;
    }
    else
    {
      HasRatRoots = 1;
      sP = -Max(-vP);
      dbprint(ppl,"// -2-4- minimal integer root found");
      dbprint(ppl-1, sP);
      //    int sP = minIntRoot(bbs,1);
//       P =  normalize(P);
//       bs = -subst(bs,s,0);
      if (sP >=0)
      {
        dbprint(ppl,"// -2-5- nonnegative root, setting s0 = -1");
        sP = -1;
      }
      else
      {
        dbprint(ppl,"// -2-5- the root is negative");
      }
    }
  }

  if (HasRatRoots)
  {
    setring @R2;
    K2 = subst(I,s,sP);
    // IF min int root exists ->
    // create the ordinary Weyl algebra and put the result into it,
    // thus creating the ring @R5
    // ELSE : return the same ring with new objects
    // keep: N, i,j,s, tmp, RL
    Nnew = Nnew - 1; // former 2*N;
    // list RL = ringlist(save);  // is defined earlier
    //  kill Lord, tmp, iv;
    list L = 0;
    list Lord, tmp;
    intvec iv;
    list RL = ringlist(basering);
    L[1] = RL[1];
    L[4] = RL[4];  //char, minpoly
    // check whether vars have admissible names -> done earlier
    // list Name = RL[2]M
    // DName is defined earlier
    list NName; // = RL[2]; // skip the last var 's'
    for (i=1; i<=Nnew; i++)
    {
      NName[i] =  RL[2][i];
    }
    L[2] = NName;
    // dp ordering;
    string s = "iv=";
    for (i=1; i<=Nnew; i++)
    {
      s = s+"1,";
    }
    s[size(s)] = ";";
    execute(s);
    tmp     = 0;
    tmp[1]  = "dp";  // string
    tmp[2]  = iv;  // intvec
    Lord[1] = tmp;
    kill s;
    tmp[1]  = "C";
    iv = 0;
    tmp[2]  = iv;
    Lord[2] = tmp;
    tmp     = 0;
    L[3]    = Lord;
    // we are done with the list
    // Add: Plural part
    def @R4@ = ring(L);
    setring @R4@;
    int N = Nnew/2;
    matrix @D[Nnew][Nnew];
    for (i=1; i<=N; i++)
    {
      @D[i,N+i]=1;
    }
    def @R4 = nc_algebra(1,@D);
    setring @R4;
    kill @R4@;
    dbprint(ppl,"// -3-1- the ring @R4 is ready");
    dbprint(ppl-1, @R4);
    ideal K4 = imap(@R2,K2);
    option(redSB);
    dbprint(ppl,"// -3-2- the final cosmetic std");
    K4 = groebner(K4);  // std does the job too
    // total cleanup
    setring @R2;
    ideal bs = imap(@R3,bs);
    bs = -normalize(bs); // "-" for getting correct coeffs!
    bs = subst(bs,s,0);
    kill @R3;
    setring @R4;
    ideal bs = imap(@R2,bs); // only rationals are the entries 
    list BS; BS[1] = bs; BS[2] = m;
    export BS;
    //    list LBS = imap(@R3,LBS);
    //    list BS; BS[1] = sbs; BS[2] = m;
    //    BS;
    //    export BS;
    ideal LD0 = K4;
    export LD0;
    return(@R4);    
  }
  else
  { 
    /* SHOULD NEVER GET THERE */
    /* no rational/integer roots */
    /* return objects in the copy of current ring */
    setring @R2;
    ideal LD0 = I;
    poly BS = normalize(K2[1]);
    export LD0;
    export BS;
    return(@R2);
  }  
}
example;
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def R = Weyl();    setring R;
  poly F = x2-y3;
  ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F; 
  def W = SDLoc(I,F);  setring W; // creates ideal LD
  def U = DLoc0(LD, x2-y3);  setring U;
  LD0; 
  BS;
}


proc SDLoc(ideal I, poly F)
"USAGE:  SDLoc(I, F);  I an ideal, F a poly
RETURN:  ring
PURPOSE: compute a generic presentation of the localization of D/I w.r.t. f^s, where D is a Weyl Algebra
ASSUME: the basering is a Weyl algebra
NOTE:    activate this ring with the @code{setring} command. In this ring,
@*       - the ideal LD (which is a Groebner basis) is the presentation of the localization
@*       If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example SDLoc; shows examples
"
{
  /* analogue to Sannfs */
  /* printlevel >=4 gives debug info */
  /* assume: we're in the Weyl algebra D  in x1,x2,...,d1,d2,... */
  def save = basering;
  /* 1. create D <t, dt, s > as in LOT */
  /* ordering: eliminate t,dt */
  int ppl = printlevel-voice+2;
  int N = nvars(save); N = N div 2;
  int Nnew = 2*N + 3; // t,Dt,s
  int i,j;
  string s;
  list RL = ringlist(save);
  list L, Lord;
  list tmp;
  intvec iv;
  L[1] = RL[1]; // char
  L[4] = RL[4]; // char, minpoly
  // check whether vars have admissible names
  list Name  = RL[2];
  list RName;
  RName[1] = "@t";
  RName[2] = "@Dt";
  RName[3] = "s";
  for(i=1;i<=N;i++)
  {
    for(j=1; j<=size(RName);j++)
    {
      if (Name[i] == RName[j])
      {
        ERROR("Variable names should not include @t,@Dt,s");
      }
    }
  }
  // now, create the names for new vars
  tmp    =  0;
  tmp[1] = "@t";
  tmp[2] = "@Dt";
  list SName ; SName[1] = "s";
  list NName = tmp + Name + SName;
  L[2]   = NName;
  tmp    = 0;
  kill NName;
  // block ord (a(1,1),dp);
  tmp[1]  = "a"; // string
  iv      = 1,1;
  tmp[2]  = iv; //intvec
  Lord[1] = tmp;
  // continue with dp 1,1,1,1...
  tmp[1]  = "dp"; // string
  s       = "iv=";
  for(i=1;i<=Nnew;i++)
  {
    s = s+"1,";
  }
  s[size(s)]= ";";
  execute(s);
  tmp[2]    = iv;
  Lord[2]   = tmp;
  tmp[1]    = "C";
  iv        = 0;
  tmp[2]    = iv;
  Lord[3]   = tmp;
  tmp       = 0;
  L[3]      = Lord;
  // we are done with the list
  def @R@ = ring(L);
  setring @R@;
  matrix @D[Nnew][Nnew];
  @D[1,2]=1;
  for(i=1; i<=N; i++)
  {
    @D[2+i,N+2+i]=1;
  }
  // ADD [s,t]=-t, [s,Dt]=Dt
  @D[1,Nnew] = -var(1);
  @D[2,Nnew] = var(2);
  def @R = nc_algebra(1,@D);
  setring @R;
  kill @R@;
  dbprint(ppl,"// -1-1- the ring @R(t,Dt,_x,_Dx,s) is ready");
  dbprint(ppl-1, @R);
  poly  F = imap(save,F);
  ideal I = imap(save,I);
  dbprint(ppl-1, "the ideal after map:");
  dbprint(ppl-1, I);
  poly p = 0;
  for(i=1; i<=N; i++)
  {
    p = diff(F,var(2+i))*@Dt + var(2+N+i);
    dbprint(ppl-1, p);
    I = subst(I,var(2+N+i),p);
    dbprint(ppl-1, var(2+N+i));
    p = 0;
  }
  I = I, @t - F;
  // t*Dt + s +1 reduced with t-f gives f*Dt + s
  I = I, F*var(2) + var(Nnew);
  // -------- the ideal I is ready ----------
  dbprint(ppl,"// -1-2- starting the elimination of @t,@Dt in @R");
  dbprint(ppl-1, I);
  //  ideal J = engine(I,eng);
  ideal J = groebner(I);
  dbprint(ppl-1,"// -1-2-1- result of the  elimination of @t,@Dt in @R");
  dbprint(ppl-1, J);;
  ideal K = nselect(J,1,2);
  dbprint(ppl,"// -1-3- @t,@Dt are eliminated");
  dbprint(ppl-1, K);  // K is without t, Dt
  K = groebner(K);  // std does the job too
  // now, we must change the ordering
  // and create a ring without t, Dt
  setring save;
  // ----------- the ring @R3 ------------
  // _x, _Dx,s;  elim.ord for _x,_Dx.
  // keep: N, i,j,s, tmp, RL
  Nnew = 2*N+1;
  kill Lord, tmp, iv, RName;
  list Lord, tmp;
  intvec iv;
  L[1] = RL[1];
  L[4] = RL[4];  // char, minpoly
  // check whether vars hava admissible names -> done earlier
  // now, create the names for new var
  tmp[1] = "s";
  list NName = Name + tmp;
  L[2] = NName;
  tmp = 0;
  // block ord (dp(N),dp);
  // string s is already defined
  s = "iv=";
  for (i=1; i<=Nnew-1; i++)
  {
    s = s+"1,";
  }
  s[size(s)]=";";
  execute(s);
  tmp[1] = "dp";  // string
  tmp[2] = iv;   // intvec
  Lord[1] = tmp;
  // continue with dp 1,1,1,1...
  tmp[1] = "dp";  // string
  s[size(s)] = ",";
  s = s+"1;";
  execute(s);
  kill s;
  kill NName;
  tmp[2]      = iv;
  Lord[2]     = tmp;
  tmp[1]      = "C";  iv  = 0;  tmp[2]=iv;
  Lord[3]     = tmp;  tmp = 0;
  L[3]        = Lord;
  // we are done with the list. Now add a Plural part
  def @R2@ = ring(L);
  setring @R2@;
  matrix @D[Nnew][Nnew];
  for (i=1; i<=N; i++)
  {
    @D[i,N+i]=1;
  }
  def @R2 = nc_algebra(1,@D);
  setring @R2;
  kill @R2@;
  dbprint(ppl,"//  -2-1- the ring @R2(_x,_Dx,s) is ready");
  dbprint(ppl-1, @R2);
  ideal MM = maxideal(1);
  MM = 0,s,MM;
  map R01 = @R, MM;
  ideal K = R01(K);
  // total cleanup
  ideal LD = K;
  // make leadcoeffs positive
  for (i=1; i<= ncols(LD); i++)
  {
    if (leadcoef(LD[i]) <0 )
    {
      LD[i] = -LD[i];
    }
  }
  export LD;
  kill @R;
  return(@R2);
}
example;
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def R = Weyl();
  setring R;
  poly F = x2-y3;
  ideal I = Dx*F, Dy*F;
  def W = SDLoc(I,F);
  setring W;
  LD;
}

proc exCusp()
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def R = Weyl();   setring R;
  poly F = x2-y3;
  ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F; 
  def W = SDLoc(I,F);
  setring W;
  LD;
  def U = DLoc0(LD,x2-y3);
  setring U;
  LD0;
  BS; 
  // the same with DLoc:
  setring R;
  DLoc(I,F);
}

proc exWalther1()
{
  // p.18 Rem 3.10
  ring r = 0,(x,Dx),dp;
  def R = nc_algebra(1,1);
  setring R;
  poly F = x;
  ideal I = x*Dx+1;
  def W = SDLoc(I,F);
  setring W;
  LD;
  ideal J = LD, x;
  eliminate(J,x*Dx); // must be [1]=s // agree!
  // the same result with Dloc0:
  def U = DLoc0(LD,x);
  setring U;
  LD0;
  BS;
}

proc exWalther2()
{
  // p.19 Rem 3.10 cont'd
  ring r = 0,(x,Dx),dp;
  def R = nc_algebra(1,1);
  setring R;
  poly F = x;
  ideal I = (x*Dx)^2+1;
  def W = SDLoc(I,F);
  setring W;
  LD;
  ideal J = LD, x;
  eliminate(J,x*Dx); // must be [1]=s^2+2*s+2 // agree!
  // the same result with Dloc0:
  def U = DLoc0(LD,x);
  setring U;
  LD0;
  BS;
  // almost the same with DLoc
  setring R;
  DLoc(I,F);
  LD0;  BS;
}

proc exWalther3()
{
  // can check with annFs too :-)
  // p.21 Ex 3.15
  LIB "nctools.lib";
  ring r = 0,(x,y,z,w,Dx,Dy,Dz,Dw),dp;
  def R = Weyl();
  setring R;
  poly F = x2+y2+z2+w2;
  ideal I = Dx,Dy,Dz,Dw;
  def W = SDLoc(I,F);
  setring W;
  LD;
  ideal J = LD, x2+y2+z2+w2;
  eliminate(J,x*y*z*w*Dx*Dy*Dz*Dw); // must be [1]=s^2+3*s+2 // agree
  ring r2 =  0,(x,y,z,w),dp;
  poly F = x2+y2+z2+w2;
  def Z = annfs(F);
  setring Z;
  LD;
  BS;
  // the same result with Dloc0:
  setring W;
  def U = DLoc0(LD,x2+y2+z2+w2);
  setring U;
  LD0;  BS;
  // the same result with DLoc:
  setring R;
  DLoc(I,F);
  LD0;  BS;
}