source: git/Singular/LIB/dmodapp.lib @ 6a07eb

//////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Noncommutative";
info="
LIBRARY: dmodapp.lib     Applications of algebraic D-modules
AUTHORS: Viktor Levandovskyy,      levandov@math.rwth-aachen.de
@*             Daniel Andres, daniel.andres@math.rwth-aachen.de

GUIDE: Let K be a field of characteristic 0, R = K[x1,..xN] and
@* D be the Weyl algebra in variables x1,..xN,d1,..dN.
@* In this library there are the following procedures for algebraic D-modules:
@* - localization of a holonomic module D/I with respect to a mult. closed set
@* of all powers of a given polynomial F from R. Our aim is to compute an
@* ideal L in D, such that D/L is a presentation of a localized module. Such L
@* always exists, since such localizations are known to be holonomic and thus
@* cyclic modules. The procedures for the localization are DLoc,SDLoc and DLoc0.
@*
@* - annihilator in D of a given polynomial F from R as well as
@* of a given rational function G/F from Quot(R). These can be computed via
@* procedures annPoly resp. annRat.
@*
@* - initial form and initial ideals in Weyl algebras with respect to a given
@* weight vector can be computed with  inForm, initialMalgrange, initialIdealW.
@*
@* - appelF1, appelF2 and appelF4 return ideals in parametric Weyl algebras,
@* which annihilate corresponding Appel hypergeometric functions.

REFERENCES:
@* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric
@*         Differential Equations', Springer, 2000
@* (ONW) Oaku, Takayama, Walther 'A Localization Algorithm for D-modules', 2000

MAIN PROCEDURES:

annPoly(f);   annihilator of a polynomial f in the corr. Weyl algebra
annRat(f,g);  annihilator of a rational function f/g in the corr. Weyl algebra
DLoc(I,F);     presentation of the localization of D/I w.r.t. f^s
SDLoc(I, F);  a generic presentation of the localization of D/I w.r.t. f^s
DLoc0(I, F);  presentation of the localization of D/I w.r.t. f^s, based on SDLoc

initialMalgrange(f[,s,t,v]); Groebner basis of the initial Malgrange ideal for f
initialIdealW(I,u,v[,s,t]); initial ideal of a given ideal w.r.t. given weights
inForm(f,w);     initial form of a poly/ideal w.r.t. a given weight
isFsat(I, F);       check whether the ideal I is F-saturated

AUXILIARY PROCEDURES:

bFactor(F);  computes the roots of irreducible factors of an univariate poly
appelF1();      create an ideal annihilating Appel F1 function
appelF2();      create an ideal annihilating Appel F2 function
appelF4();      create an ideal annihilating Appel F4 function
engine(I,i);     computes a Groebner basis with the algorithm given by i
poly2list(f);   decompose a polynomial into a list of terms and exponents
fl2poly(L,s);  reconstruct a monic univariate polynomial from its factorization
insertGenerator(id,p[,k]); insert an element into an ideal/module
deleteGenerator(id,k); delete the k-th element from an ideal/module


SEE ALSO: bfun_lib, dmod_lib, dmodvar_lib, gmssing_lib

KEYWORDS: D-module; annihilator of polynomial; annihilator of rational function; D-localization;
localization of D-module; Appel function; Appel hypergeometric function;
";

LIB "poly.lib";
LIB "sing.lib";
LIB "primdec.lib";
LIB "dmod.lib";    // loads e.g. nctools.lib
LIB "bfun.lib";    //formerly LIB "bfct.lib";
LIB "nctools.lib"; // for isWeyl etc
LIB "gkdim.lib";

// todo: complete and include into above list
// charVariety(I);       compute the characteristic variety of the ideal I
// charInfo(); ???


///////////////////////////////////////////////////////////////////////////////
// testing for consistency of the library:
proc testdmodapp()
{
  example initialIdealW;
  example initialMalgrange;
  example DLoc;
  example DLoc0;
  example SDLoc;
  example inForm;
  example isFsat;
  example annRat;
  example annPoly;
  example appelF1;
  example appelF2;
  example appelF4;
  example poly2list;
  example fl2poly;
  example insertGenerator;
  example deleteGenerator;
  example bFactor;
}

proc inForm (ideal I, intvec w)
"USAGE:  inForm(I,w);  I ideal, w intvec
RETURN:  the initial form of I w.r.t. the weight vector w
PURPOSE: computes the initial form of an ideal w.r.t. a given weight vector
NOTE:  the size of the weight vector must be equal to the number of variables
@*     of the basering.
EXAMPLE: example inForm; shows examples
"
{
  if (size(w) != nvars(basering))
  {
    ERROR("weight vector has wrong dimension");
  }
  if (I == 0)
  {
    return(I);
  }
  int j,i,s,m;
  list l;
  poly g;
  ideal J;
  for (j=1; j<=ncols(I); j++)
  {
    l = poly2list(I[j]);
    m = scalarProd(w,l[1][1]);
    g = l[1][2];
    for (i=2; i<=size(l); i++)
    {
      s = scalarProd(w,l[i][1]);
      if (s == m)
      {
        g = g + l[i][2];
      }
      else
      {
        if (s > m)
        {
          m = s;
          g = l[i][2];
        }
      }
    }
    J[j] = g;
  }
  return(J);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring @D = 0,(x,y,Dx,Dy),dp;
  def D = Weyl();
  setring D;
  poly F = 3*x^2*Dy+2*y*Dx;
  poly G = 2*x*Dx+3*y*Dy+6;
  ideal I = F,G;
  intvec w1 = -1,-1,1,1;
  intvec w2 = -1,-2,1,2;
  intvec w3 = -2,-3,2,3;
  inForm(I,w1);
  inForm(I,w2);
  inForm(I,w3);
}

/*

proc charVariety(ideal I)
"USAGE:  charVariety(I);  I an ideal
RETURN:  ring
PURPOSE: compute the characteristic variety of a D-module D/I
STATUS: experimental, todo
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 charVariety; 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;
  list ncr =ncRelations(save);
  matrix @C = ncr[1];
  matrix @D = ncr[2];
  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;
  // make @U noncommutative
  matrix @C = imap(save,@C);
  matrix @D = imap(save,@D);
  def @@U = nc_algebra(@C,@D);
  setring @@U; kill @U;
  // 2. compute Groebner basis
  ideal I = imap(save,I);
  //  I = groebner(I);
  I = slimgb(I); // a bug?
  setring @CU;
  ideal CV = imap(@@U,I);
  //  CV = groebner(CV); // cosmetics
  CV = slimgb(CV);
  export CV;
  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);
}

/*

// TODO

/*
proc charInfo(ideal I)
"USAGE:  charInfo(I);  I an ideal
RETURN:  ring
STATUS: experimental, todo
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 appelF1()
"USAGE:  appelF1();
RETURN:  ring  (and exports an ideal into it)
PURPOSE: define the ideal in a parametric Weyl algebra,
@* which annihilates Appel F1 hypergeometric function
NOTE: the ideal called  IAppel1 is exported to the output ring
EXAMPLE: example appelF1; shows examples
"
{
  // Appel F1, d = b', SST p.48
  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
  matrix @D[4][4];
  @D[1,3]=1; @D[2,4]=1;
  def @S = nc_algebra(1,@D);
  setring @S;
  ideal IAppel1 =
    (x*Dx)*(x*Dx+y*Dy+c-1) - x*(x*Dx+y*Dy+a)*(x*Dx+b),
    (y*Dy)*(x*Dx+y*Dy+c-1) - y*(x*Dx+y*Dy+a)*(y*Dy+d),
    (x-y)*Dx*Dy - d*Dx + b*Dy;
  export IAppel1;
  kill @r;
  return(@S);
}
example
{
  "EXAMPLE:"; echo = 2;
  def A = appelF1();
  setring A;
  IAppel1;
}

proc appelF2()
"USAGE:  appelF2();
RETURN:  ring (and exports an ideal into it)
PURPOSE: define the ideal in a parametric Weyl algebra,
@* which annihilates Appel F2 hypergeometric function
NOTE: the ideal called  IAppel2 is exported to the output ring
EXAMPLE: example appelF2; shows examples
"
{
  // Appel F2, c = b', SST p.85
  ring @r = (0,a,b,c),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
  matrix @D[4][4];
  @D[1,3]=1; @D[2,4]=1;
  def @S = nc_algebra(1,@D);
  setring @S;
  ideal IAppel2 =
    (x*Dx)^2 - x*(x*Dx+y*Dy+a)*(x*Dx+b),
    (y*Dy)^2 - y*(x*Dx+y*Dy+a)*(y*Dy+c);
  export IAppel2;
  kill @r;
  return(@S);
}
example
{
  "EXAMPLE:"; echo = 2;
  def A = appelF2();
  setring A;
  IAppel2;
}

proc appelF4()
"USAGE:  appelF4();
RETURN:  ring  (and exports an ideal into it)
PURPOSE: define the ideal in a parametric Weyl algebra,
@* which annihilates Appel F4 hypergeometric function
NOTE: the ideal called  IAppel4 is exported to the output ring
EXAMPLE: example appelF4; shows examples
"
{
  // Appel F4, d = c', SST, p. 39
  ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp);
  matrix @D[4][4];
  @D[1,3]=1; @D[2,4]=1;
  def @S = nc_algebra(1,@D);
  setring @S;
  ideal IAppel4 =
    Dx*(x*Dx+c-1) - (x*Dx+y*Dy+a)*(x*Dx+y*Dy+b),
    Dy*(y*Dy+d-1) - (x*Dx+y*Dy+a)*(x*Dx+y*Dy+b);
  export IAppel4;
  kill @r;
  return(@S);
}
example
{
  "EXAMPLE:"; echo = 2;
  def A = appelF4();
  setring A;
  IAppel4;
}

proc poly2list (poly f)
"USAGE:  poly2list(f); f a poly
RETURN:  list of exponents and corresponding terms of f
PURPOSE: convert a polynomial to a list of exponents and corresponding terms
EXAMPLE: example poly2list; shows examples
"
{
  list l;
  int i = 1;
  if (f == 0) // just for the zero polynomial
  {
    l[1] = list(leadexp(f), lead(f));
  }
  else { l[size(f)] = list(); } // memory pre-allocation
  while (f != 0)
  {
    l[i] = list(leadexp(f), lead(f));
    f = f - lead(f);
    i++;
  }
  return(l);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  poly F = x;
  poly2list(F);
  ring r2 = 0,(x,y),dp;
  poly F = x2y+x*y2;
  poly2list(F);
}

proc isFsat(ideal I, poly F)
"USAGE:  isFsat(I, F);  I an ideal, F a poly
RETURN: int
PURPOSE: check whether the ideal I is F-saturated
NOTE:    1 is returned if I is F-saturated, otherwise 0 is returned.
@*   we check indeed that Ker(D --F--> D/I) is (0)
EXAMPLE: example isFsat; shows examples
"
{
  /* 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 */
  if (attrib(I,"isSB")!=1)
  {
    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 (in Groebner basis) is the presentation of the localization
@* the list BS contains roots with multiplicities of Bernstein polynomial of (D/I)_f.
DISPLAY: 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 */
  if (dmodappassumeViolation())
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }
  if (!isWeyl())
  {
    ERROR("Basering is not a Weyl algebra");
  }
  if (defined(LD0) || defined(BS))
  {
    ERROR("Reserved names LD0 and/or BS are used. Please rename the objects.");
  }
  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; // Weyl algebra in variables x,y,Dx,Dy
  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;
  // I is not holonomic, since its dimension is not 4/2=2
  gkdim(I);
  DLoc(I, x2-y3); // exports LD0 and BS
  LD0; // localized module (R/I)_f is isomorphic to R/LD0
  BS; // description of b-function for localization
}

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 (in Groebner basis) is the presentation of the localization
@* the list BS contains roots and multiplicities of Bernstein polynomial of (D/I)_f.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example DLoc0; shows examples
"
{
  if (dmodappassumeViolation())
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }
  /* assume: to be run in the output ring of SDLoc */
  /* doing: 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 = ideal(matrix(lead(P))-matrix(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);
    intvec vopt = option(get);
    option(redSB);
    dbprint(ppl,"// -3-2- the final cosmetic std");
    K4 = groebner(K4);  // std does the job too
    option(set,vopt);
    // 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; // Weyl algebra in variables x,y,Dx,Dy
  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;
  // moreover I is not holonomic, since its dimension is not 2 = 4/2
  gkdim(I); // 3
  def W = SDLoc(I,F);  setring W; // creates ideal LD in W = R[s]
  def U = DLoc0(LD, x2-y3);  setring U; // compute in R
  LD0; // Groebner basis of the presentation of localization
  BS; // description of b-function for localization
}


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
ASSUME: the basering D is a Weyl algebra
NOTE:    activate this ring with the @code{setring} command. In this ring,
@*  the ideal LD (in Groebner basis) is the presentation of the localization
DISPLAY: 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,... */

  if (dmodappassumeViolation())
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }
  if (!isWeyl())
  {
    ERROR("Basering is not a Weyl algebra");
  }
  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); // @s
  // -------- 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(); // Weyl algebra on the variables x,y,Dx,Dy
  setring R;
  poly F = x2-y3;
  ideal I = Dx*F, Dy*F;
  // note, that I is not holonomic, since it's dimension is not 2
  gkdim(I); // 3, while dim R = 4
  def W = SDLoc(I,F);
  setring W; // = R[s], where s is a new variable
  LD; // Groebner basis of s-parametric presentation
}

proc annRat(poly g, poly f)
"USAGE:  annRat(g,f);  f, g polynomials
RETURN:  ring
PURPOSE: compute the annihilator of the rational function g/f in the Weyl algebra D
NOTE: activate the output ring with the @code{setring} command.
@*      In the output ring, the ideal LD (in Groebner basis) is the annihilator.
@*      The algorithm uses the computation of ann f^{-1} via D-modules.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
SEE ALSO: annPoly
EXAMPLE: example annRat; shows examples
"
{

  if (dmodappassumeViolation())
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }

  // assumptions: f is not a constant
  if (f==0) { ERROR("Denominator cannot be zero"); }
  if (leadexp(f) == 0)
  {
    // f = const, so use annPoly
    g = g/f;
    def @R = annPoly(g);
    return(@R);
  }
    // computes the annihilator of g/f
  def save = basering;
  int ppl = printlevel-voice+2;
  dbprint(ppl,"// -1-[annRat] computing the ann f^s");
  def  @R1 = SannfsBM(f);
  setring @R1;
  poly f = imap(save,f);
  int i,mir;
  int isr = 0; // checkRoot1(LD,f,1); // roots are negative, have to enter positive int
  if (!isr)
  {
    // -1 is not the root
    // find the m.i.r iteratively
    mir = 0;
    for(i=nvars(save)+1; i>=1; i--)
    {
      isr =  checkRoot1(LD,f,i);
      if (isr) { mir =-i; break; }
    }
    if (mir ==0)
    {
      "No integer root found! Aborting computations, inform the authors!";
      return(0);
    }
    // now mir == i is m.i.r.
  }
  else
  {
    // -1 is the m.i.r
    mir = -1;
  }
  dbprint(ppl,"// -2-[annRat] the minimal integer root is ");
  dbprint(ppl-1, mir);
  // use annfspecial
  dbprint(ppl,"// -3-[annRat] running annfspecial ");
  ideal AF = annfspecial(LD,f,mir,-1); // ann f^{-1}
  //  LD = subst(LD,s,j);
  //  LD = engine(LD,0);
  // modify the ring: throw s away
  // output ring comes from SannfsBM
  list U = ringlist(@R1);
  list tmp; // variables
  for(i=1; i<=size(U[2])-1; i++)
  {
    tmp[i] = U[2][i];
  }
  U[2] = tmp;
  tmp = 0;
  tmp[1] = U[3][1]; // x,Dx block
  tmp[2] = U[3][3]; // module block
  U[3] = tmp;
  tmp = 0;
  tmp = U[1],U[2],U[3],U[4];
  def @R2 = ring(tmp);
  setring @R2;
  // now supply with Weyl algebra relations
  int N = nvars(@R2)/2;
  matrix @D[2*N][2*N];
  for(i=1; i<=N; i++)
  {
    @D[i,N+i]=1;
  }
  def @R3 = nc_algebra(1,@D);
  setring @R3;
  dbprint(ppl,"// - -[annRat] ring without s is ready:");
  dbprint(ppl-1,@R3);
  poly g = imap(save,g);
  matrix G[1][1] = g;
  matrix LL = matrix(imap(@R1,AF));
  kill @R1;   kill @R2;
  dbprint(ppl,"// -4-[annRat] running modulo");
  ideal LD  = modulo(G,LL);
  dbprint(ppl,"// -4-[annRat] running GB on the final result");
  LD  = engine(LD,0);
  export LD;
  return(@R3);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly g = 2*x*y;  poly f = x^2 - y^3;
  def B = annRat(g,f);
  setring B;
  LD;
  // Now, compare with the output of Macaulay2:
  ideal tst = 3*x*Dx + 2*y*Dy + 1, y^3*Dy^2 - x^2*Dy^2 + 6*y^2*Dy + 6*y,
9*y^2*Dx^2*Dy-4*y*Dy^3+27*y*Dx^2+2*Dy^2, 9*y^3*Dx^2-4*y^2*Dy^2+10*y*Dy -10;
 option(redSB); option(redTail);
 LD = groebner(LD);
 tst = groebner(tst);
 print(matrix(NF(LD,tst)));  print(matrix(NF(tst,LD)));
 // So, these two answers are the same
}

/*
//static proc ex_annRat()
{
  // more complicated example for annRat
  ring r = 0,(x,y,z),dp;
  poly f = x3+y3+z3; // mir = -2
  poly g = x*y*z;
  def A = annRat(g,f);
  setring A;
}
*/

proc annPoly(poly f)
"USAGE:  annPoly(f);  f a poly
RETURN:  ring
PURPOSE: compute the complete annihilator ideal of f in the Weyl algebra D
NOTE:  activate the output ring with the @code{setring} command.
@*   In the output ring, the ideal LD (in Groebner basis) is the annihilator.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
SEE ALSO: annRat
EXAMPLE: example annPoly; shows examples
"
{
  // computes a system of linear PDEs with polynomial coeffs for f
  def save = basering;
  list L = ringlist(save);
  list Name = L[2];
  int N = nvars(save);
  int i;
  for (i=1; i<=N; i++)
  {
    Name[N+i] = "D"+Name[i]; // concat
  }
  L[2] = Name;
  def @R = ring(L);
  setring @R;
  def @@R = Weyl();
  setring @@R;
  kill @R;
  matrix M[1][N];
  for (i=1; i<=N; i++)
  {
    M[1,i] = var(N+i);
  }
  matrix F[1][1] = imap(save,f);
  ideal I = modulo(F,M);
  ideal LD = groebner(I);
  export LD;
  return(@@R);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),dp;
  poly f = x^2*z - y^3;
  def A = annPoly(f);
  setring A; // A is the 3rd Weyl algebra in 6 variables
  LD; // the Groebner basis of annihilator
  gkdim(LD); // must be 3 = 6/2, since A/LD is holonomic module
  NF(Dy^4, LD); // must be 0 since Dy^4 clearly annihilates f
}

/* DIFFERENT EXAMPLES

//static 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);
}

//static 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;
}

//static 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;
}

//static 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;
}

*/

proc engine(def I, int i)
"USAGE:  engine(I,i);  I  ideal/module/matrix, i an int
RETURN:  the same type as I
PURPOSE: compute the Groebner basis of I with the algorithm, chosen via i
NOTE: By default and if i=0, slimgb is used; otherwise std does the job.
EXAMPLE: example engine; shows examples
"
{
  /* std - slimgb mix */
  def J;
  //  ideal J;
  if (i==0)
  {
    J = slimgb(I);
  }
  else
  {
    // without options -> strange! (ringlist?)
    intvec v = option(get);
    option(redSB);
    option(redTail);
    J = std(I);
    option(set, v);
  }
  return(J);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),Dp;
  ideal I  = y*(x3-y2),x*(x3-y2);
  engine(I,0); // uses slimgb
  engine(I,1); // uses std
}

proc insertGenerator (list #)
"USAGE:  insertGenerator(id,p[,k]);  id an ideal/module, p a poly/vector, k an optional int
RETURN:  same as id
PURPOSE: inserts p into the first argument at k-th index position and returns the enlarged object
NOTE:    If k is given, p is inserted at position k, otherwise (and by default),
@*       p is inserted at the beginning.
EXAMPLE: example insertGenerator; shows examples
"
{
  if (size(#) < 2)
  {
    ERROR("insertGenerator has to be called with at least 2 arguments (ideal/module,poly/vector)");
  }
  string inp1 = typeof(#[1]);
  if (inp1 == "ideal" || inp1 == "module")
  {
    if (inp1 == "ideal") { ideal id = #[1]; }
    else { module id = #[1]; }
  }
  else { ERROR("first argument has to be of type ideal or module"); }
  string inp2 = typeof(#[2]);
  if (inp2 == "poly" || inp2 == "vector")
  {
    if (inp2 =="poly") { poly f = #[2]; }
    else
    {
      if (inp1 == "ideal")
      {
        ERROR("second argument has to be a polynomial if first argument is an ideal");
      }
      else { vector f = #[2]; }
    }
  }
  else { ERROR("second argument has to be of type poly/vector"); }
  int n = ncols(id);
  int k = 1; // default
  if (size(#)>=3)
  {
    if (typeof(#[3]) == "int")
    {
      k = #[3];
      if (k<=0)
      {
        ERROR("third argument has to be positive");
      }
    }
    else { ERROR("third argument has to be of type int"); }
  }
  execute(inp1 +" J;");
  if (k == 1) { J = f,id; }
  else
  {
    if (k>n)
    {
      J = id;
      J[k] = f;
    }
    else // 1<k<=n
    {
      J[1..k-1] = id[1..k-1];
      J[k] = f;
      J[k+1..n+1] = id[k..n];
    }
  }
  return(J);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),dp;
  ideal I = x^2,z^4;
  insertGenerator(I,y^3);
  insertGenerator(I,y^3,2);
  module M = I;
  insertGenerator(M,[x^3,y^2,z],2);
}

proc deleteGenerator (list #)
"USAGE:  deleteGenerator(id,k);  id an ideal/module, k an int
RETURN:  same as id
PURPOSE: deletes the k-th generator from the first argument and returns the altered object
EXAMPLE: example insertGenerator; shows examples
"
{
  if (size(#) < 2)
  {
    ERROR("deleteGenerator has to be called with 2 arguments (ideal/module,int)");
  }
  string inp1 = typeof(#[1]);
  if (inp1 == "ideal" || inp1 == "module")
  {
    if (inp1 == "ideal") { ideal id = #[1]; }
    else { module id = #[1]; }
  }
  else { ERROR("first argument has to be of type ideal or module"); }
  string inp2 = typeof(#[2]);
  if (inp2 == "int" || inp2 == "number") { int k = int(#[2]); }
  else { ERROR("second argument has to be of type int"); }
  int n = ncols(id);
  if (n == 1) { ERROR(inp1+" must have more than one generator"); }
  if (k<=0 || k>n) { ERROR("second argument has to be in the range 1,...,"+string(n)); }
  execute(inp1 +" J;");
  if (k == 1) { J = id[2..n]; }
  else
  {
    if (k == n) { J = id[1..n-1]; }
    else
    {
      J[1..k-1] = id[1..k-1];
      J[k..n-1] = id[k+1..n];
    }
  }
  return(J);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),dp;
  ideal I = x^2,y^3,z^4;
  deleteGenerator(I,2);
  module M = [x,y,z],[x2,y2,z2],[x3,y3,z3];;
  deleteGenerator(M,2);
}

proc fl2poly(list L, string s)
"USAGE:  fl2poly(L,s); L a list, s a string
RETURN:  poly
PURPOSE: reconstruct a monic polynomial in one variable from its factorization
ASSUME:  s is a string with the name of some variable and
@*         L is supposed to consist of two entries:
@*         L[1] of the type ideal with the roots of a polynomial
@*         L[2] of the type intvec with the multiplicities of corr. roots
EXAMPLE: example fl2poly; shows examples
"
{
  if (varNum(s)==0)
  {
    ERROR("no such variable found in the basering"); return(0);
  }
  poly x = var(varNum(s));
  poly P = 1;
  int sl = size(L[1]);
  ideal RR = L[1];
  intvec IV = L[2];
  for(int i=1; i<= sl; i++)
  {
    if (IV[i] > 0)
    {
      P = P*((x-RR[i])^IV[i]);
    }
    else
    {
      printf("Ignored the root with incorrect multiplicity %s",string(IV[i]));
    }
  }
  return(P);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z,s),Dp;
  ideal I = -1,-4/3,-5/3,-2;
  intvec mI = 2,1,1,1;
  list BS = I,mI;
  poly p = fl2poly(BS,"s");
  p;
  factorize(p,2);
}

static proc safeVarName (string s, string cv)
{
  string S;
  if (cv == "v")  { S = "," + "," + varstr(basering)  + ","; }
  if (cv == "c")  { S = "," + "," + charstr(basering) + ","; }
  if (cv == "cv") { S = "," + charstr(basering) + "," + varstr(basering) + ","; }
  s = "," + s + ",";
  while (find(S,s) <> 0)
  {
    s[1] = "@";
    s = "," + s;
  }
  s = s[2..size(s)-1];
  return(s)
}

proc initialIdealW (ideal I, intvec u, intvec v, list #)
"USAGE:  initialIdealW(I,u,v [,s,t,w]);  I ideal, u,v intvecs, s,t optional ints,
@*       w an optional intvec
RETURN:  ideal, GB of initial ideal of the input ideal w.r.t. the weights u and v
ASSUME:  The basering is the n-th Weyl algebra in characteristic 0 and  for all
@*       1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 holds, i.e. the
@*       sequence of variables is given by x(1),...,x(n),D(1),...,D(n),
@*       where D(i) is the differential operator belonging to x(i).
PURPOSE: computes the initial ideal with respect to given weights.
NOTE:    u and v are understood as weight vectors for x(1..n) and D(1..n)
@*       respectively.
@*       If s<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       If t<>0, a matrix ordering is used for Groebner basis computations,
@*       otherwise, and by default, a block ordering is used.
@*       If w consist of 2n strictly positive entries, w is used for weighted
@*       homogenization, otherwise, and by default, no weights are used.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example initialIdealW; shows examples
"
{
  if (dmodappassumeViolation())
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }
  if (!isWeyl())
  {
    ERROR("Basering is not a Weyl algebra.");
  }
  int ppl = printlevel - voice +2;
  def save = basering;
  int n = nvars(save)/2;
  int N = 2*n+1;
  list RL = ringlist(save);
  int i;
  int whichengine = 0;           // default
  int methodord   = 0;           // default
  intvec homogweights = 1:(2*n); // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      whichengine = int(#[1]);
    }
    if (size(#)>1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        methodord = int(#[2]);
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="intvec")
        {
          if (size(#[3])==2*n && allPositive(#[3])==1)
          {
            homogweights = #[3];
          }
          else
          {
            print("// Homogenization weight vector must consist of positive entries and be");
            print("// of size " + string(n) + ". Using no weights.");
          }
        }
      }
    }
  }
  for (i=1; i<=n; i++)
  {
    if (bracket(var(i+n),var(i))<>1)
    {
      ERROR(string(var(i+n)) + " is not a differential operator for " + string(var(i)));
    }
  }
  // 1. create  homogenized Weyl algebra
  // 1.1 create ordering
  intvec uv = u,v,0;
  homogweights = homogweights,1;
  list Lord = list(list("a",homogweights));
  list C0 = list("C",intvec(0));
  if (methodord == 0) // default: blockordering
  {
    Lord[2] = list("a",uv);
    Lord[3] = list("dp",intvec(1:(N-1)));
    Lord[4] = list("lp",intvec(1));
    Lord[5] = C0;
  }
  else // M() ordering
  {
    intmat @Ord[N][N];
    @Ord[1,1..N] = uv; @Ord[2,1..N] = 1:(N-1);
    for (i=1; i<=N-2; i++) { @Ord[2+i,N - i] = -1; }
    dbprint(ppl-1,"// the ordering matrix:",@Ord);
    Lord[2] = list("M",intvec(@Ord));
    Lord[3] = C0;
  }
  // 1.2 the new var
  list Lvar = RL[2]; Lvar[N] = safeVarName("h","cv");
  // 1.3 create commutative ring
  list L@@Dh = RL; L@@Dh = L@@Dh[1..4];
  L@@Dh[2] = Lvar; L@@Dh[3] = Lord;
  def @Dh = ring(L@@Dh); kill L@@Dh;
  setring @Dh;
  dbprint(ppl-1,"// the ring @Dh:",@Dh);
  // 1.4 create non-commutative relations
  matrix @relD[N][N];
  for (i=1; i<=n; i++) { @relD[i,n+i] = var(N)^(homogweights[i]+homogweights[n+i]); }
  def Dh = nc_algebra(1,@relD);
  setring Dh; kill @Dh;        
  dbprint(ppl-1,"// computing in ring",Dh);
  // 2. Compute the initial ideal
  ideal I = imap(save,I);
  I = homog(I,h);
  // 2.1 the hard part: Groebner basis computation
  dbprint(ppl, "// starting Groebner basis computation with engine: "+string(whichengine));
  I = engine(I, whichengine);
  dbprint(ppl, "// finished Groebner basis computation");
  dbprint(ppl-1, "// I before dehomogenization is " +string(I));
  I = subst(I,var(N),1); // dehomogenization
  dbprint(ppl-1, "I after dehomogenization is " +string(I));
  // 2.2 the initial form
  I = inForm(I,uv);
  setring save;
  I = imap(Dh,I); kill Dh;
  // 2.3 the final GB
  dbprint(ppl, "// starting cosmetic Groebner basis computation with engine: "+string(whichengine));
  I = engine(I, whichengine);
  dbprint(ppl,"// finished cosmetic Groebner basis computation");
  return(I);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring @D = 0,(x,Dx),dp;
  def D = Weyl();
  setring D;
  intvec u = -1; intvec v = 2;
  ideal I = x^2*Dx^2,x*Dx^4;
  ideal J = initialIdealW(I,u,v); J;
}

proc initialMalgrange (poly f,list #)
"USAGE:  initialMalgrange(f,[,a,b,v]); f poly, a,b optional ints, v opt. intvec
RETURN:  ring, Weyl algebra induced by basering, extended by two new vars t,Dt
PURPOSE: computes the initial Malgrange ideal of a given polynomial w.r.t. the weight
@*       vector (-1,0...,0,1,0,...,0) such that the weight of t is -1 and the
@*       weight of Dt is 1.
ASSUME:  The basering is commutative and of characteristic 0.
NOTE:    Activate the output ring with the @code{setring} command.
@*       The returned ring contains the ideal \"inF\", being the initial ideal
@*       of the Malgrange ideal of f.
@*       Varnames of the basering should not include t and Dt.
@*       If a<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       If b<>0, a matrix ordering is used for Groebner basis computations,
@*       otherwise, and by default, a block ordering is used.
@*       If a positive weight vector v is given, the weight
@*       (d,v[1],...,v[n],1,d+1-v[1],...,d+1-v[n]) is used for homogenization
@*       computations, where d denotes the weighted degree of f.
@*       Otherwise and by default, v is set to (1,...,1). See Noro, 2002.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example initialMalgrange; shows examples
"
{
  if (dmodappassumeViolation())
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }
  if (!isCommutative())
  {
    ERROR("Basering must be commutative.");
  }
  int ppl = printlevel - voice +2;
  def save = basering;
  int n = nvars(save);
  int i;
  int whichengine = 0; // default
  int methodord   = 0; // default
  intvec u0 = 1:n;     // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      whichengine = int(#[1]);
    }
    if (size(#)>1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        methodord = int(#[2]);
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="intvec" && size(#[3])==n && allPositive(#[3])==1)
        {
          u0 = #[3];
        }
      }
    }
  }
  // creating the homogenized extended Weyl algebra
  list RL = ringlist(save);
  RL = RL[1..4]; // if basering is commutative nc_algebra
  list C0 = list("C",intvec(0));
  // 1. create homogenization weights
  // 1.1. get the weighted degree of f
  list Lord = list(list("wp",u0),C0);
  list L@r = RL;
  L@r[3] = Lord;
  def r = ring(L@r); kill L@r,Lord;
  setring r;
  poly f = imap(save,f);
  int d = deg(f);
  setring save; kill r;
  // 1.2 the homogenization weights
  intvec homogweights = d;
  homogweights[n+2] = 1;
  for (i=1; i<=n; i++)
  {
    homogweights[i+1]   = u0[i];
    homogweights[n+2+i] = d+1-u0[i];
  }
  // 2. create extended Weyl algebra
  int N = 2*n+2;
  // 2.1 create names for vars
  string vart  = safeVarName("t","cv");
  string varDt = safeVarName("D"+vart,"cv");
  while (varDt <> "D"+vart)
  {
    vart  = safeVarName("@"+vart,"cv");
    varDt = safeVarName("D"+vart,"cv");
  }
  list Lvar;
  Lvar[1] = vart; Lvar[n+2] = varDt;
  for (i=1; i<=n; i++)
  {
    Lvar[i+1]   = string(var(i));
    Lvar[i+n+2] = safeVarName("D" + string(var(i)),"cv");
  }
  //  2.2 create ordering
  list Lord = list(list("dp",intvec(1:N)),C0);
  // 2.3 create the (n+1)-th Weyl algebra
  list L@D = RL; L@D[2] = Lvar; L@D[3] = Lord;
  def @D = ring(L@D); kill L@D;
  setring @D;
  def D = Weyl();
  setring D; kill @D;
  dbprint(ppl,"// the (n+1)-th Weyl algebra :" ,D);
  // 3. compute the initial ideal
  // 3.1 create the Malgrange ideal
  poly f = imap(save,f);
  ideal I = var(1)-f;
  for (i=1; i<=n; i++)
  {
    I = I, var(n+2+i)+diff(f,var(i+1))*var(n+2);
  }
  // I = engine(I,whichengine); // todo is it efficient to compute a GB first wrt dp first?
  // 3.2 computie the initial ideal
  intvec w = 1,0:n;
  I = initialIdealW(I,-w,w,whichengine,methodord,homogweights);
  ideal inF = I; attrib(inF,"isSB",1);
  export(inF);
  setring save;
  return(D);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly f = x^2+y^3+x*y^2;
  def D = initialMalgrange(f);
  setring D;
  inF;
  setring r;
  intvec v = 3,2;
  def D2 = initialMalgrange(f,1,1,v);
  setring D2;
  inF;
}

proc dmodappassumeViolation()
{
  // returns Boolean : yes/no [for assume violation]
  // char K = 0
  // no qring
  if (  (size(ideal(basering)) >0) || (char(basering) >0) )
  {
    //    "ERROR: no qring is allowed";
    return(1);
  }
  return(0);
}

proc bFactor (poly F)
"USAGE:  bFactor(f);  f poly
RETURN:  list
PURPOSE: tries to compute the roots of a univariate poly f
NOTE:    The output list consists of two or three entries:
@*       roots of f as an ideal, their multiplicities as intvec, and,
@*       if present, a third one being the product of all irreducible factors
@*       of degree greater than one, given as string.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example bFactor; shows examples
"
{
  int ppl = printlevel - voice +2;
  def save = basering;
  ideal LI = variables(F);
  list L;
  int i = size(LI);
  if (i>1) { ERROR("poly has to be univariate")}
  if (i == 0)
  {
    dbprint(ppl,"// poly is constant");
    L = list(ideal(0),intvec(0),string(F));
    return(L);
  }
  poly v = LI[1];
  L = ringlist(save); L = L[1..4];
  L[2] = list(string(v));
  L[3] = list(list("dp",intvec(1)),list("C",intvec(0)));
  def @S = ring(L);
  setring @S;
  poly ir = 1; poly F = imap(save,F);
  list L = factorize(F);
  ideal I = L[1];
  intvec m = L[2];
  ideal II; intvec mm;
  for (i=2; i<=ncols(I); i++)
  {
    if (deg(I[i]) > 1)
    {
      ir = ir * I[i]^m[i];
    }
    else
    {
      II[size(II)+1] = I[i];
      mm[size(II)] = m[i];
    }
  }
  II = normalize(II);
  II = subst(II,var(1),0);
  II = -II;
  if (size(II)>0)
  {
    dbprint(ppl,"// found roots");
    dbprint(ppl-1,string(II));
  }
  else
  {
    dbprint(ppl,"// found no roots");
  }
  L = list(II,mm);
  if (ir <> 1)
  {
    dbprint(ppl,"// found irreducible factors");
    ir = cleardenom(ir);
    dbprint(ppl-1,string(ir));
    L = L + list(string(ir));
  }
  else
  {
    dbprint(ppl,"// no irreducible factors found");
  }
  setring save;
  L = imap(@S,L);
  return(L);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  bFactor((x^2-1)^2);
  bFactor((x^2+1)^2);
  bFactor((y^2+1/2)*(y+9)*(y-7));
}