source: git/Singular/LIB/dmodapp.lib @ 858cae

//////////////////////////////////////////////////////////////////////////////
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

Support: DFG Graduiertenkolleg 1632 'Experimentelle und konstruktive Algebra'

OVERVIEW:
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:

@* - given a cyclic representation D/I of a holonomic module and a polynomial
 F in R, it is proved that the localization of D/I with respect to the mult.
 closed set of all powers of F is a holonomic D-module. Thus we aim to compute
 its cyclic representaion D/L for an ideal L in D. 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.

@* - Groebner bases with respect to weights (according to (SST), given an
 arbitrary integer vector containing weights for variables, one computes the
 homogenization of a given ideal relative to this vector, then one computes a
 Groebner basis and returns the dehomogenization of the result), initial
 forms and initial ideals in Weyl algebras with respect to a given weight
 vector can be computed with GBWeight, inForm, initialMalgrange and
 initialIdealW.

@* - restriction and integration of a holonomic module D/I. Suppose I
 annihilates a function F(x1,...,xn). Our aim is to compute an ideal J
 directly from I, which annihilates
@*   - F(0,...,0,xk,...,xn) in case of restriction or
@*   - the integral of F with respect to x1,...,xm in case of integration.
 The corresponding procedures are restrictionModule, restrictionIdeal,
 integralModule and integralIdeal.

@* - characteristic varieties defined by ideals in Weyl algebras can be computed
 with charVariety and charInfo.

@* - 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
@* (OTW) Oaku, Takayama, Walther 'A Localization Algorithm for D-modules',
         Journal of Symbolic Computation, 2000
@* (OT)  Oaku, Takayama 'Algorithms for D-modules',
         Journal of Pure and Applied Algebra, 1998


PROCEDURES:

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

GBWeight(I,u,v[,a,b]);       computes Groebner basis of I wrt a weight vector
initialMalgrange(f[,s,t,v]); computes Groebner basis of initial Malgrange ideal
initialIdealW(I,u,v[,s,t]);  computes initial ideal of  wrt a given weight
inForm(f,w);                 computes initial form of poly/ideal wrt a weight

restrictionIdeal(I,w[,eng,m,G]);  computes restriction ideal of I wrt w
restrictionModule(I,w[,eng,m,G]); computes restriction module of I wrt w
integralIdeal(I,w[,eng,m,G]);     computes integral ideal of I wrt w
integralModule(I,w[,eng,m,G]);    computes integral module of I wrt w
deRhamCohom(f[,eng,m]);           computes basis of n-th de Rham cohom. group
deRhamCohomIdeal(I[,w,eng,m,G]);  computes basis of n-th de Rham cohom. group

charVariety(I); computes characteristic variety of the ideal I
charInfo(I);    computes char. variety, singular locus and primary decomp.
isFsat(I,F);    checks whether the ideal I is F-saturated



appelF1();     creates an ideal annihilating Appel F1 function
appelF2();     creates an ideal annihilating Appel F2 function
appelF4();     creates an ideal annihilating Appel F4 function

fourier(I[,v]);        applies Fourier automorphism to ideal
inverseFourier(I[,v]); applies inverse Fourier automorphism to ideal

bFactor(F);    computes the roots of irreducible factors of an univariate poly
intRoots(L);   dismisses non-integer roots from list in bFactor format
poly2list(f);  decomposes the polynomial f into a list of terms and exponents
fl2poly(L,s);  reconstructs a monic univariate polynomial from its factorization

insertGenerator(id,p[,k]); inserts an element into an ideal/module
deleteGenerator(id,k);     deletes the k-th element from an ideal/module

engine(I,i);   computes a Groebner basis with the algorithm specified by i
isInt(n);      checks whether number n is actually an int
sortIntvec(v); sorts intvec

KEYWORDS: D-module; annihilator of polynomial; annihilator of rational function;
D-localization; localization of D-module; D-restriction; restriction of
D-module; D-integration; integration of D-module; characteristic variety;
Appel function; Appel hypergeometric function

SEE ALSO: bfun_lib, dmod_lib, dmodvar_lib, gmssing_lib
";

/*
  Changelog
  21.09.10 by DA:
  - restructured library for better readability
  - new / improved procs:
  - toolbox: isInt, intRoots, sortIntvec
  - GB wrt weights: GBWeight, initialIdealW rewritten using GBWeight
  - restriction/integration: restrictionX, integralX where X in {Module, Ideal},
  fourier, inverseFourier, deRhamCohom, deRhamCohomIdeal
  - characteristic variety: charVariety, charInfo
  - added keywords for features
  - reformated help strings and examples in existing procs
  - added SUPPORT in header
  - added reference (OT)

  04.10.10 by DA:
  - incorporated suggestions by Oleksandr Motsak, among other:
  - bugfixes for fl2poly, sortIntvec, annPoly, GBWeight
  - enhanced functionality for deleteGenerator, inForm

  11.10.10 by DA:
  - procedure bFactor now sorts the roots using new static procedure sortNumberIdeal

  17.03.11 by DA:
  - bugfixes for inForm with polynomial input, typo in restrictionIdealEngine
*/


LIB "bfun.lib";     // for pIntersect etc
LIB "dmod.lib";     // for SannfsBM etc
LIB "general.lib";  // for sort
LIB "gkdim.lib";
LIB "nctools.lib";  // for isWeyl etc
LIB "poly.lib";
LIB "primdec.lib";  // for primdecGKZ
LIB "qhmoduli.lib"; // for Max
LIB "sing.lib";     // for slocus


///////////////////////////////////////////////////////////////////////////////
// testing for consistency of the library:
proc testdmodapp()
{
  example annPoly;
  example annRat;
  example DLoc;
  example SDLoc;
  example DLoc0;
  example GBWeight;
  example initialMalgrange;
  example initialIdealW;
  example inForm;
  example restrictionIdeal;
  example restrictionModule;
  example integralIdeal;
  example integralModule;
  example deRhamCohom;
  example deRhamCohomIdeal;
  example charVariety;
  example charInfo;
  example isFsat;
  example appelF1;
  example appelF2;
  example appelF4;
  example fourier;
  example inverseFourier;
  example bFactor;
  example intRoots;
  example poly2list;
  example fl2poly;
  example insertGenerator;
  example deleteGenerator;
  example engine;
  example isInt;
  example sortIntvec;
}


// general assumptions ////////////////////////////////////////////////////////

static proc dmodappAssumeViolation()
{
  // char K <> 0 or qring
  if (  (size(ideal(basering)) >0) || (char(basering) >0) )
  {
    ERROR("Basering is inappropriate: characteristic>0 or qring present");
  }
  return();
}

static proc dmodappMoreAssumeViolation()
{
  // char K <> 0, qring
  dmodappAssumeViolation();
  // no Weyl algebra
  if (isWeyl() == 0)
  {
    ERROR("Basering is not a Weyl algebra");
  }
  // wrong sequence of vars
  int i,n;
  n = nvars(basering)/2;
  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)));
    }
  }
  return();
}

static proc safeVarName (string s, string cv)
// assumes 's' to be a valid variable name
// returns valid var name string @@..@s
{
  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)
    }

static proc intLike (def i)
{
  string str = typeof(i);
  if (str == "int" || str == "number" || str == "bigint")
  {
    return(1);
  }
  else
  {
    return(0);
  }
}


// toolbox ////////////////////////////////////////////////////////////////////

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 poly2list (poly f)
  "USAGE:  poly2list(f); f a poly
RETURN:  list of exponents and corresponding terms of f
PURPOSE: converts a poly to a list of pairs consisting of intvecs (1st entry)
@*       and polys (2nd entry), where the i-th pair contains the exponent of the
@*       i-th term of f and the i-th term (with coefficient) itself.
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+5xy2;
  poly2list(F);
  poly2list(0);
}

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(s+ " is no variable in the basering");
  }
  poly x = var(varNum(s));
  poly P = 1;
  ideal RR = L[1];
  int sl = ncols(RR);
  intvec IV = L[2];
  if (sl <> nrows(IV))
  {
    ERROR("number of roots doesn't match number of multiplicites");
  }
  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,-4/3,0,-5/3,-2;
  intvec mI = 2,1,2,1,1;
  list BS = I,mI;
  poly p = fl2poly(BS,"s");
  p;
  factorize(p,2);
}

proc insertGenerator (list #)
  "USAGE:  insertGenerator(id,p[,k]);
@*       id an ideal/module, p a poly/vector, k an optional int
RETURN:  of the same type as id
PURPOSE: inserts p into id at k-th 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 (k=1).
SEE ALSO: deleteGenerator
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")
  {
    def id = #[1];
  }
  else { ERROR("first argument has to be of type ideal or module"); }
  string inp2 = typeof(#[2]);
  if (inp2 == "poly" || inp2 == "vector")
  {
    def f = #[2];
  }
  else { ERROR("second argument has to be of type poly/vector"); }
  if (inp1 == "ideal" && inp2 == "vector")
  {
    ERROR("second argument has to be a polynomial if first argument is an ideal");
  }
  // don't check module/poly combination due to auto-conversion
  //   if (inp1 == "module" && inp2 == "poly")
  //   {
  //     ERROR("second argument has to be a vector if first argument is a module");
  //   }
  int n = ncols(id);
  int k = 1; // default
  if (size(#)>=3)
  {
    if (intLike(#[3]))
    {
      k = int(#[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[n+1] = id[n]; // preinit
      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*gen(3);
  insertGenerator(M,[x^3,y^2,z],2);
  insertGenerator(M,x+y+z,4);
}

proc deleteGenerator (def id, int k)
  "USAGE:   deleteGenerator(id,k);  id an ideal/module, k an int
RETURN:   of the same type as id
PURPOSE:  deletes the k-th generator from the first argument and returns
@*        the altered object
SEE ALSO: insertGenerator
EXAMPLE:  example deleteGenerator; shows examples
"
{
  string inp1 = typeof(id);
  if (inp1 <> "ideal" && inp1 <> "module")
  {
    ERROR("first argument has to be of type ideal or module");
  }
  execute(inp1 +" J;");
  int n = ncols(id);
  if (n == 1 && k == 1) { return(J); }
  if (k<=0 || k>n)
  {
    ERROR("second argument has to be in the range 1,...,"+string(n));
  }
  J[n-1] = 0; // preinit
  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];
  print(deleteGenerator(M,2));
  M = M[1];
  deleteGenerator(M,1);
}

static proc sortNumberIdeal (ideal I)
// sorts ideal of constant polys (ie numbers), returns according permutation
{
  int i;
  int nI = ncols(I);
  intvec dI;
  for (i=nI; i>0; i--)
  {
    dI[i] = int(denominator(leadcoef(I[i])));
  }
  int lcmI = lcm(dI);
  for (i=nI; i>0; i--)
  {
    dI[i] = int(lcmI*leadcoef(I[i]));
  }
  intvec perm = sortIntvec(dI)[2];
  return(perm);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,s,dp;
  ideal I = -9/20,-11/20,-23/20,-19/20,-1,-13/10,-27/20,-13/20,-21/20,-17/20,
    -11/10,-9/10,-7/10; // roots of BS poly of reiffen(4,5)
  intvec v = sortNumberIdeal(I); v;
  I[v];
}

proc bFactor (poly F)
  "USAGE:  bFactor(f);  f poly
RETURN:  list of ideal and intvec and possibly a string
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.
@*       If f is the zero polynomial or if f has no roots in the ground field,
@*       this is encoded as root 0 with multiplicity 0.
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;
  intvec perm = sortNumberIdeal(II);
  II = II[perm];
  mm = mm[perm];
  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));
  bFactor(1);
  bFactor(0);
}

proc isInt (number n)
  "USAGE:  isInt(n); n a number
RETURN:  int, 1 if n is an integer or 0 otherwise
PURPOSE: check whether given object of type number is actually an int
NOTE:    Parameters are treated as integers.
EXAMPLE: example isInt; shows an example
"
{
  number d = denominator(n);
  if (d<>1)
  {
    return(0);
  }
  else
  {
    return(1);
  }
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  number n = 4/3;
  isInt(n);
  n = 11;
  isInt(n);
}

proc intRoots (list l)
  "USAGE:  isInt(L); L a list
RETURN:  list
PURPOSE: extracts integer roots from a list given in @code{bFactor} format
ASSUME:  The input list must be given in the format of @code{bFactor}.
NOTE:    Parameters are treated as integers.
SEE ALSO: bFactor
EXAMPLE: example intRoots; shows an example
"
{
  int wronginput;
  int sl = size(l);
  if (sl>0)
  {
    if (typeof(l[1])<>"ideal"){wronginput = 1;}
    if (sl>1)
    {
      if (typeof(l[2])<>"intvec"){wronginput = 1;}
      if (sl>2)
      {
        if (typeof(l[3])<>"string"){wronginput = 1;}
        if (sl>3){wronginput = 1;}
      }
    }
  }
  if (sl<2){wronginput = 1;}
  if (wronginput)
  {
    ERROR("Given list has wrong format.");
  }
  int i,j;
  ideal l1 = l[1];
  int n = ncols(l1);
  j = 1;
  ideal I;
  intvec v;
  for (i=1; i<=n; i++)
  {
    if (size(l1[j])>1) // poly not number
    {
      ERROR("Ideal in list has wrong format.");
    }
    if (isInt(leadcoef(l1[i])))
    {
      I[j] = l1[i];
      v[j] = l[2][i];
      j++;
    }
  }
  return(list(I,v));
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  list L = bFactor((x-4/3)*(x+3)^2*(x-5)^4); L;
  intRoots(L);
}

proc sortIntvec (intvec v)
  "USAGE:  sortIntvec(v); v an intvec
RETURN:  list of two intvecs
PURPOSE: sorts an intvec
NOTE:    In the output list L, the first entry consists of the entries of v
@*       satisfying L[1][i] >= L[1][i+1]. The second entry is a permutation
@*       such that v[L[2]] = L[1].
@*       Unlike in the procedure @code{sort}, zeros are not dismissed.
SEE ALSO: sort
EXAMPLE: example sortIntvec; shows examples
"
{
  int i;
  intvec vpos,vzero,vneg,vv,sortv,permv;
  list l;
  for (i=1; i<=nrows(v); i++)
  {
    if (v[i]>0)
    {
      vpos = vpos,i;
    }
    else
    {
      if (v[i]==0)
      {
        vzero = vzero,i;
      }
      else // v[i]<0
      {
        vneg = vneg,i;
      }
    }
  }
  if (size(vpos)>1)
  {
    vpos = vpos[2..size(vpos)];
    vv = v[vpos];
    l = sort(vv);
    vv = l[1];
    vpos = vpos[l[2]];
    sortv = vv[size(vv)..1];
    permv = vpos[size(vv)..1];
  }
  if (size(vzero)>1)
  {
    vzero = vzero[2..size(vzero)];
    permv = permv,vzero;
    sortv = sortv,0:size(vzero);
  }
  if (size(vneg)>1)
  {
    vneg = vneg[2..size(vneg)];
    vv = -v[vneg];
    l = sort(vv);
    vv = -l[1];
    vneg = vneg[l[2]];
    sortv = sortv,vv;
    permv = permv,vneg;
  }
  if (permv[1]==0)
  {
    sortv = sortv[2..size(sortv)];
    permv = permv[2..size(permv)];
  }
  return(list(sortv,permv));
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  intvec v = -1,0,1,-2,0,2;
  list L = sortIntvec(v); L;
  v[L[2]];
  v = -3,0;
  sortIntvec(v);
  v = 0,-3;
  sortIntvec(v);
}


// F-saturation ///////////////////////////////////////////////////////////////

proc isFsat(ideal I, poly F)
  "USAGE:  isFsat(I, F);  I an ideal, F a poly
RETURN:  int, 1  if I is F-saturated and 0 otherwise
PURPOSE: checks whether the ideal I is F-saturated
NOTE:    We check indeed that Ker(D--> F--> D/I) is 0, where D is the basering.
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 I is F-sat */
  if (attrib(I,"isSB")!=1)
  {
    I = groebner(I);
  }
  matrix @M = matrix(I);
  matrix @F[1][1] = F;
  def S = modulo(module(@F),module(@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);
}


// annihilators ///////////////////////////////////////////////////////////////

proc annRat(poly g, poly f)
  "USAGE:   annRat(g,f);  f, g polynomials
RETURN:   ring (a Weyl algebra) containing an ideal 'LD'
PURPOSE:  compute the annihilator of the rational function g/f in the
@*        corresponding Weyl algebra
ASSUME:   basering is commutative and over a field of characteristic 0
NOTE:     Activate the output ring with the @code{setring} command.
@*        In the output ring, the ideal 'LD' (in Groebner basis) is the
@*        annihilator of g/f.
@*        The algorithm uses the computation of Ann(f^{-1}) via D-modules,
@*        see (SST).
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
"
{
  // assumption check
  dmodappAssumeViolation();
  if (!isCommutative())
  {
    ERROR("Basering must be commutative.");
  }
  // assumptions: f is not a constant
  if (f==0) { ERROR("the denominator f cannot be zero"); }
  if ((leadexp(f) == 0) && (size(f) < 2))
  {
    // 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)
    {
      ERROR("No integer root found! Aborting computations, inform the authors!");
    }
    // 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
}

proc annPoly(poly f)
  "USAGE:   annPoly(f);  f a poly
RETURN:   ring (a Weyl algebra) containing an ideal 'LD'
PURPOSE:  compute the complete annihilator ideal of f in the corresponding
@*        Weyl algebra
ASSUME:   basering is commutative and over a field of characteristic 0
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
"
{
  // assumption check
  dmodappAssumeViolation();
  if (!isCommutative())
  {
    ERROR("Basering must be commutative.");
  }
  // 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] = safeVarName("D"+Name[i],"cv"); // 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);
  def I = modulo(module(F),module(M));
  ideal LD = I;
  LD = groebner(LD);
  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
  poly f = imap(r,f);
  NF(LD*f,std(ideal(Dx,Dy,Dz))); // must be zero if LD indeed annihilates f
}



// localizations //////////////////////////////////////////////////////////////

proc DLoc(ideal I, poly F)
  "USAGE:  DLoc(I, f);  I an ideal, f a poly
RETURN:  list of ideal and list
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 output list L,
@*       - L[1] is an ideal (given as Groebner basis), the presentation of the
@*       localization,
@*       - L[2] is a list containing 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 */
  dmodappAssumeViolation();
  if (!isWeyl())
  {
    ERROR("Basering is not a 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);
  ideal bs = imap(@R3,bs);
  list BS; BS[1] = bs; BS[2] = m;
  kill @R3;
  printlevel = old_printlevel;
  return(list(LD0,BS));
}
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);
  list L = DLoc(I, x2-y3);
  L[1]; // localized module (R/I)_f is isomorphic to R/LD0
  L[2]; // description of b-function for localization
}

proc DLoc0(ideal I, poly F)
  "USAGE:  DLoc0(I, f);  I an ideal, f a poly
RETURN:  ring (a Weyl algebra) containing an ideal 'LD0' and a list 'BS'
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 the output ring with the @code{setring} command. In this ring,
@*       - the ideal LD0 (given as 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
"
{
  dmodappAssumeViolation();
  /* 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 (basering extended by a new variable) containing an ideal 'LD'
PURPOSE: compute a generic presentation of the localization of D/I w.r.t. f^s
ASSUME:  the basering D is a Weyl algebra over a field of characteristic 0
NOTE:    Activate this ring with the @code{setring} command. In this ring,
@*       the ideal LD (given as 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,... */

  dmodappAssumeViolation();
  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
}


// Groebner basis wrt weights and initial ideal business //////////////////////

proc GBWeight (ideal I, intvec u, intvec v, list #)
  "USAGE:  GBWeight(I,u,v [,s,t,w]);
@*       I ideal, u,v intvecs, s,t optional ints, w an optional intvec
RETURN:  ideal, Groebner basis of I w.r.t. the weights u and v
ASSUME:  The basering is the n-th Weyl algebra over a field of 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 a Groebner basis with respect to given weights
NOTE:    The weights u and v are understood as weight vectors for x(i) and D(i),
@*       respectively. According to (SST), one computes the homogenization of a
@*       given ideal relative to (u,v), then one computes a Groebner basis and
@*       returns the dehomogenization of the result.
@*       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 is given and consists of exactly 2*n strictly positive entries,
@*       w is used for constructing the weighted homogenized Weyl algebra,
@*       see Noro (2002). Otherwise, and by default, the homogenization weight
@*       (1,...,1) is used.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example GBWeight; shows examples
"
{
  dmodappMoreAssumeViolation();
  int ppl = printlevel - voice +2;
  def save = basering;
  int n = nvars(save)/2;
  int whichengine = 0;           // default
  int methodord   = 0;           // default
  intvec homogweights = 1:(2*n); // default
  if (size(#)>0)
  {
    if (intLike(#[1]))
    {
      whichengine = int(#[1]);
    }
    if (size(#)>1)
    {
      if (intLike(#[2]))
      {
        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 weight (1,...,1).");
          }
        }
      }
    }
  }
  // 1. create  homogenized Weyl algebra
  // 1.1 create ordering
  int i;
  list RL = ringlist(save);
  int N = 2*n+1;
  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[5] = C0;
    Lord[4] = list("lp",intvec(1));
    Lord[3] = list("dp",intvec(1:(N-1)));
    Lord[2] = list("a",uv);
  }
  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 homog 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;
  // 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,var(N));
  // 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, "// ideal before dehomogenization is " +string(I));
  I = subst(I,var(N),1); // dehomogenization
  setring save;
  I = imap(Dh,I); kill Dh;
  return(I);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def D2 = Weyl();
  setring D2;
  ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6;
  intvec u = -2,-3;
  intvec v = -u;
  GBWeight(I,u,v);
  ideal J = std(I);
  GBWeight(J,u,v); // same as above
  u = 0,1;
  GBWeight(I,u,v);
}

proc inForm (def I, intvec w)
"USAGE:  inForm(I,w);  I ideal or poly, w intvec
RETURN:  ideal, generated by initial forms of generators of I w.r.t. w, or
@*       poly, initial form of input poly w.r.t. w
PURPOSE: computes the initial form of an ideal or a poly w.r.t. the weight w
NOTE:    The size of the weight vector must be equal to the number of variables
@*       of the basering.
EXAMPLE: example inForm; shows examples
"
{
  string inp1 = typeof(I);
  if ((inp1 <> "ideal") && (inp1 <> "poly"))
  {
    ERROR("first argument has to be an ideal or a poly");
  }
  if (size(w) != nvars(basering))
  {
    ERROR("weight vector has wrong dimension");
  }
  ideal II = I;
  if (simplify(II,2) == 0)
  {
    return(I);
  }
  int j,i;
  bigint s,m;
  list l;
  poly g;
  ideal J;
  for (j=1; j<=ncols(II); j++)
  {
    l = poly2list(II[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;
  }
  if (inp1 == "ideal")
  {
    return(J);
  }
  else
  {
    return(J[1]);
  }
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 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);
  inForm(F,w1);
}

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 wrt 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 is given and consists of exactly 2*n strictly positive entries,
@*       w is used as homogenization weight.
@*       Otherwise, and by default, the homogenization weight (1,...,1) is 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
"
{
  // assumption check in GBWeight
  int ppl = printlevel - voice + 2;
  printlevel = printlevel + 1;
  I = GBWeight(I,u,v,#);
  printlevel = printlevel - 1;
  intvec uv = u,v;
  I = inForm(I,uv);
  int eng;
  if (size(#)>0)
  {
    if(typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      eng = int(#[1]);
    }
  }
  dbprint(ppl,"// starting cosmetic Groebner basis computation");
  I = engine(I,eng);
  dbprint(ppl,"// finished cosmetic Groebner basis computation");
  return(I);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def D2 = Weyl();
  setring D2;
  ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6;
  intvec u = -2,-3;
  intvec v = -u;
  initialIdealW(I,u,v);
  ideal J = std(I);
  initialIdealW(J,u,v); // same as above
  u = 0,1;
  initialIdealW(I,u,v);
}

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 over a field 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
"
{
  dmodappAssumeViolation();
  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 (intLike(#[1]))
    {
      whichengine = int(#[1]);
    }
    if (size(#)>1)
    {
      if (intLike(#[2]))
      {
        methodord = int(#[2]);
      }
      if (size(#)>2)
      {
        if ((typeof(#[3])=="intvec") && (size(#[3])==n) && (allPositive(#[3])==1))
        {
          u0 = #[3];
        }
      }
    }
  }
  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 efficient to compute GB wrt dp first?
  // 3.2 computie the initial ideal
  intvec w = 1,0:n;
  printlevel = printlevel + 1;
  I = initialIdealW(I,-w,w,whichengine,methodord,homogweights);
  printlevel = printlevel - 1;
  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;
}


// restriction and integration ////////////////////////////////////////////////

static proc restrictionModuleEngine (ideal I, intvec w, list #)
// returns list L with 2 entries of type ideal
// L[1]=basis of free module, L[2]=generating system of submodule
// #=eng,m,G; eng=engine; m=min int root of bfctIdeal(I,w); G=GB of I wrt (-w,w)
{
  dmodappMoreAssumeViolation();
  if (!isHolonomic(I))
  {
    ERROR("Given ideal is not holonomic");
  }
  int l0,l0set,Gset;
  ideal G;
  int whichengine = 0;         // default
  if (size(#)>0)
  {
    if (intLike(#[1]))
    {
      whichengine = int(#[1]);
    }
    if (size(#)>1)
    {
      if (intLike(#[2]))
      {
        l0 = int(#[2]);
        l0set = 1;
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="ideal")
        {
          G = #[3];
          Gset = 1;
        }
      }
    }
  }
  int ppl = printlevel;
  int i,j,k;
  int n = nvars(basering)/2;
  if (w == 0:size(w))
  {
    ERROR("weight vector must not be zero");
  }
  if (size(w)<>n)
  {
    ERROR("weight vector must have exactly " + string(n) + " entries");
  }
  for (i=1; i<=n; i++)
  {
    if (w[i]<0)
    {
      ERROR("weight vector must not have negative entries");
    }
  }
  intvec ww = -w,w;
  if (!Gset)
  {
    G = GBWeight(I,-w,w,whichengine);
    dbprint(ppl,"// found GB wrt weight " +string(w));
    dbprint(ppl-1,"// " + string(G));
  }
  if (!l0set)
  {
    ideal inG = inForm(G,ww);
    inG = engine(inG,whichengine);
    poly s = 0;
    for (i=1; i<=n; i++)
    {
      s = s + w[i]*var(i)*var(i+n);
    }
    vector v = pIntersect(s,inG);
    list L = bFactor(vec2poly(v));
    dbprint(ppl,"// found b-function of given ideal wrt given weight");
    dbprint(ppl-1,"// roots: "+string(L[1]));
    dbprint(ppl-1,"// multiplicities: "+string(L[2]));
    kill inG,v,s;
    L = intRoots(L);           // integral roots of b-function
    if (L[2]==0:size(L[2]))       // no integral roots
    {
      return(list(ideal(0),ideal(0)));
    }
    intvec v;
    for (i=1; i<=ncols(L[1]); i++)
    {
      v[i] = int(L[1][i]);
    }
    l0 = Max(v);
    dbprint(ppl,"// maximal integral root is " +string(l0));
    kill L,v;
  }
  if (l0 < 0)                  // maximal integral root is < 0
  {
    return(list(ideal(0),ideal(0)));
  }
  intvec m;
  for (i=ncols(G); i>0; i--)
  {
    m[i] = deg(G[i],ww);
  }
  dbprint(ppl,"// weighted degree of generators of GB is " +string(m));
  def save = basering;
  list RL = ringlist(save);
  RL = RL[1..4];
  list Lvar;
  j = 1;
  intvec neww;
  for (i=1; i<=n; i++)
  {
    if (w[i]>0)
    {
      Lvar[j] = string(var(i+n));
      neww[j] = w[i];
      j++;
    }
  }
  list Lord;
  Lord[1] = list("dp",intvec(1:n));
  Lord[2] = list("C", intvec(0));
  RL[2] = Lvar;
  RL[3] = Lord;
  def r = ring(RL);
  kill Lvar, Lord, RL;
  setring r;
  ideal B;
  list Blist;
  intvec mm = l0,-m+l0;
  for (i=0; i<=Max(mm); i++)
  {
    B = weightKB(std(0),i,neww);
    Blist[i+1] = B;
  }
  setring save;
  list Blist = imap(r,Blist);
  ideal ff = maxideal(1);
  for (i=1; i<=n; i++)
  {
    if (w[i]<>0)
    {
      ff[i] = 0;
    }
  }
  map f = save,ff;
  ideal B,M;
  poly p;
  for (i=1; i<=size(G); i++)
  {
    for (j=1; j<=l0-m[i]+1; j++)
    {
      B = Blist[j];
      for (k=1; k<=ncols(B); k++)
      {
        p = B[k]*G[i];
        p = f(p);
        M[size(M)+1] = p;
      }
    }
  }
  ideal Bl0 = Blist[1..(l0+1)];
  dbprint(ppl,"// found basis of free module");
  dbprint(ppl-1,"// " + string(Bl0));
  dbprint(ppl,"// found generators of submodule");
  dbprint(ppl-1,"// " + string(M));
  return(list(Bl0,M));
}

static proc restrictionModuleOutput (ideal B, ideal N, intvec w, int eng, string str)
// returns ring, which contains module "str"
{
  int n = nvars(basering)/2;
  int i,j;
  def save = basering;
  // 1: create new ring
  list RL = ringlist(save);
  RL = RL[1..4];
  list V = RL[2];
  poly prodvar = 1;
  int zeropresent;
  j = 0;
  for (i=1; i<=n; i++)
  {
    if (w[i]<>0)
    {
      V = delete(V,i-j);
      V = delete(V,i-2*j-1+n);
      j = j+1;
      prodvar = prodvar*var(i)*var(i+n);
    }
    else
    {
      zeropresent = 1;
    }
  }
  if (!zeropresent) // restrict/integrate all vars, return input ring
  {
    def newR = save;
  }
  else
  {
    RL[2] = V;
    V = list();
    V[1] = list("C", intvec(0));
    V[2] = list("dp",intvec(1:(2*size(ideal(w)))));
    RL[3] = V;
    def @D = ring(RL);
    setring @D;
    def newR = Weyl();
    setring save;
    kill @D;
  }
  // 2. get coker representation of module
  module M = coeffs(N,B,prodvar);
  if (zeropresent)
  {
    setring newR;
    module M = imap(save,M);
  }
  M = engine(M,eng);
  M = prune(M);
  M = engine(M,eng);
  execute("module " + str + " = M;");
  execute("export(" + str + ");");
  setring save;
  return(newR);
}

proc restrictionModule (ideal I, intvec w, list #)
  "USAGE:  restrictionModule(I,w,[,eng,m,G]);
@*       I ideal, w intvec, eng and m optional ints, G optional ideal
RETURN:  ring (a Weyl algebra) containing a module 'resMod'
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
@*       Further, assume that I is holonomic and that w is n-dimensional with
@*       non-negative entries.
PURPOSE: computes the restriction module of a holonomic ideal to the subspace
@*       defined by the variables corresponding to the non-zero entries of the
@*       given intvec
NOTE:    The output ring is the Weyl algebra defined by the zero entries of w.
@*       It contains a module 'resMod' being the restriction module of I wrt w.
@*       If there are no zero entries, the input ring is returned.
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       The minimal integer root of the b-function of I wrt the weight (-w,w)
@*       can be specified via the optional argument m.
@*       The optional argument G is used for specifying a Groebner Basis of I
@*       wrt the weight (-w,w), that is, the initial form of G generates the
@*       initial ideal of I wrt the weight (-w,w).
@*       Further note, that the assumptions on m and G (if given) are not
@*       checked.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example restrictionModule; shows examples
"
{
  list L = restrictionModuleEngine(I,w,#);
  int eng;
  if (size(#)>0)
  {
    if (intLike(#[1]))
    {
      eng = int(#[1]);
    }
  }
  def newR = restrictionModuleOutput(L[1],L[2],w,eng,"resMod");
  return(newR);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(a,x,b,Da,Dx,Db),dp;
  def D3 = Weyl();
  setring D3;
  ideal I = a*Db-Dx+2*Da, x*Db-Da, x*Da+a*Da+b*Db+1,
    x*Dx-2*x*Da-a*Da, b*Db^2+Dx*Da-Da^2+Db,
    a*Dx*Da+2*x*Da^2+a*Da^2+b*Dx*Db+Dx+2*Da;
  intvec w = 1,0,0;
  def rm = restrictionModule(I,w);
  setring rm; rm;
  print(resMod);
}

static proc restrictionIdealEngine (ideal I, intvec w, string cf, list #)
{
  int eng;
  if (size(#)>0)
  {
    if(intLike(#[1]))
    {
      eng = int(#[1]);
    }
  }
  def save = basering;
  if (cf == "restriction")
  {
    def newR = restrictionModule(I,w,#);
    setring newR;
    matrix M = resMod;
    kill resMod;
  }
  if (cf == "integral")
  {
    def newR = integralModule(I,w,#);
    setring newR;
    matrix M = intMod;
    kill intMod;
  }
  int i,r,c;
  r = nrows(M);
  c = ncols(M);
  ideal J;
  if (r == 1) // nothing to do
  {
    J = M;
  }
  else
  {
    matrix zm[r-1][1]; // zero matrix
    matrix v[r-1][1];
    for (i=1; i<=c; i++)
    {
      if (M[1,i]<>0)
      {
        v = M[2..r,i];
        if (v == zm)
        {
          J[size(J)+1] = M[1,i];
        }
      }
    }
  }
  J = engine(J,eng);
  if (cf == "restriction")
  {
    ideal resIdeal = J;
    export(resIdeal);
  }
  if (cf == "integral")
  {
    ideal intIdeal = J;
    export(intIdeal);
  }
  setring save;
  return(newR);
}

proc restrictionIdeal (ideal I, intvec w, list #)
  "USAGE:  restrictionIdeal(I,w,[,eng,m,G]);
@*       I ideal, w intvec, eng and m optional ints, G optional ideal
RETURN:  ring (a Weyl algebra) containing an ideal 'resIdeal'
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
@*       Further, assume that I is holonomic and that w is n-dimensional with
@*       non-negative entries.
PURPOSE: computes the restriction ideal of a holonomic ideal to the subspace
@*       defined by the variables corresponding to the non-zero entries of the
@*       given intvec
NOTE:    The output ring is the Weyl algebra defined by the zero entries of w.
@*       It contains an ideal 'resIdeal' being the restriction ideal of I wrt w.
@*       If there are no zero entries, the input ring is returned.
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       The minimal integer root of the b-function of I wrt the weight (-w,w)
@*       can be specified via the optional argument m.
@*       The optional argument G is used for specifying a Groebner basis of I
@*       wrt the weight (-w,w), that is, the initial form of G generates the
@*       initial ideal of I wrt the weight (-w,w).
@*       Further note, that the assumptions on m and G (if given) are not
@*       checked.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example restrictionIdeal; shows examples
"
{
  def rm = restrictionIdealEngine(I,w,"restriction",#);
  return(rm);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(a,x,b,Da,Dx,Db),dp;
  def D3 = Weyl();
  setring D3;
  ideal I = a*Db-Dx+2*Da,
    x*Db-Da,
    x*Da+a*Da+b*Db+1,
    x*Dx-2*x*Da-a*Da,
    b*Db^2+Dx*Da-Da^2+Db,
    a*Dx*Da+2*x*Da^2+a*Da^2+b*Dx*Db+Dx+2*Da;
  intvec w = 1,0,0;
  def D2 = restrictionIdeal(I,w);
  setring D2; D2;
  resIdeal;
}

proc fourier (ideal I, list #)
  "USAGE:  fourier(I[,v]); I an ideal, v an optional intvec
RETURN:  ideal
PURPOSE: computes the Fourier transform of an ideal in a Weyl algebra
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
NOTE:    The Fourier automorphism is defined by mapping x(i) to -D(i) and
@*       D(i) to x(i).
@*       If v is an intvec with entries ranging from 1 to n, the Fourier
@*       transform of I restricted to the variables given by v is computed.
SEE ALSO: inverseFourier
EXAMPLE: example fourier; shows examples
"
{
  dmodappMoreAssumeViolation();
  intvec v;
  if (size(#)>0)
  {
    if(typeof(#[1])=="intvec")
    {
      v = #[1];
    }
  }
  int n = nvars(basering)/2;
  int i;
  if(v <> 0:size(v))
  {
    v = sortIntvec(v)[1];
    for (i=1; i<size(v); i++)
    {
      if (v[i] == v[i+1])
      {
        ERROR("No double entries allowed in intvec");
      }
    }
  }
  else
  {
    v = 1..n;
  }
  ideal m = maxideal(1);
  for (i=1; i<=size(v); i++)
  {
    if (v[i]<0 || v[i]>n)
    {
      ERROR("Entries of intvec must range from 1 to "+string(n));
    }
    m[v[i]] = -var(v[i]+n);
    m[v[i]+n] = var(v[i]);
  }
  map F = basering,m;
  ideal FI = F(I);
  return(FI);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def D2 = Weyl();
  setring D2;
  ideal I = x*Dx+2*y*Dy+2, x^2*Dx+y*Dx+2*x;
  intvec v = 2;
  fourier(I,v);
  fourier(I);
}

proc inverseFourier (ideal I, list #)
  "USAGE:  inverseFourier(I[,v]); I an ideal, v an optional intvec
RETURN:  ideal
PURPOSE: computes the inverse Fourier transform of an ideal in a Weyl algebra
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
NOTE:    The Fourier automorphism is defined by mapping x(i) to -D(i) and
@*       D(i) to x(i).
@*       If v is an intvec with entries ranging from 1 to n, the inverse Fourier
@*       transform of I restricted to the variables given by v is computed.
SEE ALSO: fourier
EXAMPLE: example inverseFourier; shows examples
"
{
  dmodappMoreAssumeViolation();
  intvec v;
  if (size(#)>0)
  {
    if(typeof(#[1])=="intvec")
    {
      v = #[1];
    }
  }
  int n = nvars(basering)/2;
  int i;
  if(v <> 0:size(v))
  {
    v = sortIntvec(v)[1];
    for (i=1; i<size(v); i++)
    {
      if (v[i] == v[i+1])
      {
        ERROR("No double entries allowed in intvec");
      }
    }
  }
  else
  {
    v = 1..n;
  }
  ideal m = maxideal(1);
  for (i=1; i<=size(v); i++)
  {
    if (v[i]<0 || v[i]>n)
    {
      ERROR("Entries of intvec must range between 1 and "+string(n));
    }
    m[v[i]] = var(v[i]+n);
    m[v[i]+n] = -var(v[i]);
  }
  map F = basering,m;
  ideal FI = F(I);
  return(FI);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,Dx,Dy),dp;
  def D2 = Weyl();
  setring D2;
  ideal I = x*Dx+2*y*Dy+2, x^2*Dx+y*Dx+2*x;
  intvec v = 2;
  ideal FI = fourier(I);
  inverseFourier(FI);
}

proc integralModule (ideal I, intvec w, list #)
  "USAGE:  integralModule(I,w,[,eng,m,G]);
@*       I ideal, w intvec, eng and m optional ints, G optional ideal
RETURN:  ring (a Weyl algebra) containing a module 'intMod'
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
@*       Further, assume that I is holonomic and that w is n-dimensional with
@*       non-negative entries.
PURPOSE: computes the integral module of a holonomic ideal w.r.t. the subspace
@*       defined by the variables corresponding to the non-zero entries of the
@*       given intvec
NOTE:    The output ring is the Weyl algebra defined by the zero entries of w.
@*       It contains a module 'intMod' being the integral module of I wrt w.
@*       If there are no zero entries, the input ring is returned.
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       Let F(I) denote the Fourier transform of I w.r.t. w.
@*       The minimal integer root of the b-function of F(I) w.r.t. the weight
@*       (-w,w) can be specified via the optional argument m.
@*       The optional argument G is used for specifying a Groebner Basis of F(I)
@*       wrt the weight (-w,w), that is, the initial form of G generates the
@*       initial ideal of F(I) w.r.t. the weight (-w,w).
@*       Further note, that the assumptions on m and G (if given) are not
@*       checked.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example integralModule; shows examples
"
{
  int l0,l0set,Gset;
  ideal G;
  int whichengine = 0;         // default
  if (size(#)>0)
  {
    if (intLike(#[1]))
    {
      whichengine = int(#[1]);
    }
    if (size(#)>1)
    {
      if (intLike(#[2]))
      {
        l0 = int(#[2]);
        l0set = 1;
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="ideal")
        {
          G = #[3];
          Gset = 1;
        }
      }
    }
  }
  int ppl = printlevel;
  int i;
  int n = nvars(basering)/2;
  intvec v;
  for (i=1; i<=n; i++)
  {
    if (w[i]>0)
    {
      if (v == 0:size(v))
      {
        v[1] = i;
      }
      else
      {
        v[size(v)+1] = i;
      }
    }
  }
  ideal FI = fourier(I,v);
  dbprint(ppl,"// computed Fourier transform of given ideal");
  dbprint(ppl-1,"// " + string(FI));
  list L;
  if (l0set)
  {
    if (Gset) // l0 and G given
    {
      L = restrictionModuleEngine(FI,w,whichengine,l0,G);
    }
    else      // l0 given, G not
    {
      L = restrictionModuleEngine(FI,w,whichengine,l0);
    }
  }
  else        // nothing given
  {
    L = restrictionModuleEngine(FI,w,whichengine);
  }
  ideal B,N;
  B = inverseFourier(L[1],v);
  N = inverseFourier(L[2],v);
  def newR = restrictionModuleOutput(B,N,w,whichengine,"intMod");
  return(newR);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,b,Dx,Db),dp;
  def D2 = Weyl();
  setring D2;
  ideal I = x*Dx+2*b*Db+2, x^2*Dx+b*Dx+2*x;
  intvec w = 1,0;
  def im = integralModule(I,w);
  setring im; im;
  print(intMod);
}

proc integralIdeal (ideal I, intvec w, list #)
  "USAGE:  integralIdeal(I,w,[,eng,m,G]);
@*       I ideal, w intvec, eng and m optional ints, G optional ideal
RETURN:  ring (a Weyl algebra) containing an ideal 'intIdeal'
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
@*       Further, assume that I is holonomic and that w is n-dimensional with
@*       non-negative entries.
PURPOSE: computes the integral ideal of a holonomic ideal w.r.t. the subspace
@*       defined by the variables corresponding to the non-zero entries of the
@*       given intvec.
NOTE:    The output ring is the Weyl algebra defined by the zero entries of w.
@*       It contains ideal 'intIdeal' being the integral ideal of I w.r.t. w.
@*       If there are no zero entries, the input ring is returned.
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       The minimal integer root of the b-function of I wrt the weight (-w,w)
@*       can be specified via the optional argument m.
@*       The optional argument G is used for specifying a Groebner basis of I
@*       wrt the weight (-w,w), that is, the initial form of G generates the
@*       initial ideal of I wrt the weight (-w,w).
@*       Further note, that the assumptions on m and G (if given) are not
@*       checked.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example integralIdeal; shows examples
"
{
  def im = restrictionIdealEngine(I,w,"integral",#);
  return(im);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,b,Dx,Db),dp;
  def D2 = Weyl();
  setring D2;
  ideal I = x*Dx+2*b*Db+2, x^2*Dx+b*Dx+2*x;
  intvec w = 1,0;
  def D1 = integralIdeal(I,w);
  setring D1; D1;
  intIdeal;
}

proc deRhamCohomIdeal (ideal I, list #)
  "USAGE:  deRhamCohomIdeal (I[,w,eng,k,G]);
@*       I ideal, w optional intvec, eng and k optional ints, G optional ideal
RETURN:  ideal
ASSUME:  The basering is the n-th Weyl algebra D over a field of characteristic
@*       zero 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).
@*       Further, assume that I is of special kind, namely let f in K[x] and
@*       consider the module K[x,1/f]f^m, where m is smaller than or equal to
@*       the minimal integer root of the Bernstein-Sato polynomial of f.
@*       Since this module is known to be a holonomic D-module, it has a cyclic
@*       presentation D/I.
PURPOSE: computes a basis of the n-th de Rham cohomology group of the complement
@*       of the hypersurface defined by f
NOTE:    If I does not satisfy the assumptions described above, the result might
@*       have no meaning. Note that I can be computed with @code{annfs}.
@*       If w is an intvec with exactly n strictly positive entries, w is used
@*       in the computation. Otherwise, and by default, w is set to (1,...,1).
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       Let F(I) denote the Fourier transform of I wrt w.
@*       An integer smaller than or equal to the minimal integer root of the
@*       b-function of F(I) wrt the weight (-w,w) can be specified via the
@*       optional argument k.
@*       The optional argument G is used for specifying a Groebner Basis of F(I)
@*       wrt the weight (-w,w), that is, the initial form of G generates the
@*       initial ideal of F(I) wrt the weight (-w,w).
@*       Further note, that the assumptions on I, k and G (if given) are not
@*       checked.
THEORY:  (SST) pp. 232-235
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
SEE ALSO: deRhamCohom
EXAMPLE: example deRhamCohomIdeal; shows examples
"
{
  intvec w = 1:(nvars(basering)/2);
  int l0,l0set,Gset;
  ideal G;
  int whichengine = 0;         // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="intvec")
    {
      if (allPositive(#[1])==1)
      {
        w = #[1];
      }
      else
      {
        print("// Entries of intvec must be strictly positive");
        print("// Using weight " + string(w));
      }
      if (size(#)>1)
      {
        if (intLike(#[2]))
        {
          whichengine = int(#[2]);
        }
        if (size(#)>2)
        {
          if (intLike(#[3]))
          {
            l0 = int(#[3]);
            l0set = 1;
          }
          if (size(#)>3)
          {
            if (typeof(#[4])=="ideal")
            {
              G = #[4];
              Gset = 1;
            }
          }
        }
      }
    }
  }
  int ppl = printlevel;
  int i,j;
  int n = nvars(basering)/2;
  intvec v;
  for (i=1; i<=n; i++)
  {
    if (w[i]>0)
    {
      if (v == 0:size(v))
      {
        v[1] = i;
      }
      else
      {
        v[size(v)+1] = i;
      }
    }
  }
  ideal FI = fourier(I,v);
  dbprint(ppl,"// computed Fourier transform of given ideal");
  dbprint(ppl-1,"// " + string(FI));
  list L;
  if (l0set)
  {
    if (Gset) // l0 and G given
    {
      L = restrictionModuleEngine(FI,w,whichengine,l0,G);
    }
    else      // l0 given, G not
    {
      L = restrictionModuleEngine(FI,w,whichengine,l0);
    }
  }
  else        // nothing given
  {
    L = restrictionModuleEngine(FI,w,whichengine);
  }
  ideal B,N;
  B = inverseFourier(L[1],v);
  N = inverseFourier(L[2],v);
  dbprint(ppl,"// computed integral module of given ideal");
  dbprint(ppl-1,"// " + string(B));
  dbprint(ppl-1,"// " + string(N));
  ideal DR;
  poly p;
  poly Dt = 1;
  for (i=1; i<=n; i++)
  {
    Dt = Dt*var(i+n);
  }
  N = simplify(N,2+8);
  printlevel = printlevel-1;
  N = linReduceIdeal(N);
  N = simplify(N,2+8);
  for (i=1; i<=size(B); i++)
  {
    p = linReduce(B[i],N);
    if (p<>0)
    {
      DR[size(DR)+1] = B[i]*Dt;
      j=1;
      while (p<N[j])
      {
        j++;
      }
      N = insertGenerator(N,p,j+1);
    }
  }
  printlevel = printlevel + 1;
  return(DR);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),dp;
  poly F = x^3+y^3+z^3;
  bfctAnn(F);            // Bernstein-Sato poly of F has minimal integer root -2
  def W = annRat(1,F^2); // so we compute the annihilator of 1/F^2
  setring W; W;          // Weyl algebra, contains LD = Ann(1/F^2)
  LD;                    // K[x,y,z,1/F]F^(-2) is isomorphic to W/LD as W-module
  deRhamCohomIdeal(LD);  // we see that the K-dim is 2
}

proc deRhamCohom (poly f, list #)
  "USAGE:  deRhamCohom(f[,eng,m]);  f poly, eng and m optional ints
RETURN:  ring (a Weyl Algebra) containing an ideal 'DR'
ASSUME:  Basering is a commutative and over a field of characteristic 0.
PURPOSE: computes a basis of the n-th de Rham cohomology group of the complement
@*       of the hypersurface defined by f, where n denotes the number of
@*       variables of the basering
NOTE:    The output ring is the n-th Weyl algebra. It contains an ideal 'DR'
@*       being a basis of the n-th de Rham cohomology group of the complement of
@*       the hypersurface defined by f.
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
@*       If m is given, it is assumed to be less than or equal to the minimal
@*       integer root of the Bernstein-Sato polynomial of f. This assumption is
@*       not checked. If not specified, m is set to the minimal integer root of
@*       the Bernstein-Sato polynomial of f.
THEORY:  (SST) pp. 232-235
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
SEE ALSO: deRhamCohomIdeal
EXAMPLE: example deRhamCohom; shows example
"
{
  int ppl = printlevel - voice + 2;
  int eng,l0,l0given;
  if (size(#)>0)
  {
    if(intLike(#[1]))
    {
      eng = int(#[1]);
    }
    if (size(#)>1)
    {
      if(intLike(#[2]))
      {
        l0 = int(#[2]);
        l0given = 1;
      }
    }
  }
  if (!isCommutative())
  {
    ERROR("Basering must be commutative.");
  }
  int i;
  def save = basering;
  int n = nvars(save);
  dbprint(ppl,"// Computing s-parametric annihilator Ann(f^s)...");
  def A = Sannfs(f);
  setring A;
  dbprint(ppl,"// ...done");
  dbprint(ppl-1,"//    Got: " + string(LD));
  poly f = imap(save,f);
  if (!l0given)
  {
    dbprint(ppl,"// Computing b-function of given poly...");
    ideal LDf = LD,f;
    LDf = engine(LDf,eng);
    vector v = pIntersect(var(2*n+1),LDf);   // BS poly of f
    list BS = bFactor(vec2poly(v));
    dbprint(ppl,"// ...done");
    dbprint(ppl-1,"// roots: " + string(BS[1]));
    dbprint(ppl-1,"// multiplicities: " + string(BS[2]));
    BS = intRoots(BS);
    intvec iv;
    for (i=1; i<=ncols(BS[1]); i++)
    {
      iv[i] = int(BS[1][i]);
    }
    l0 = Min(iv);
    kill v,iv,BS,LDf;
  }
  dbprint(ppl,"// Computing Ann(f^" + string(l0) + ")...");
  LD = annfspecial(LD,f,l0,l0); // Ann(f^l0)
  // create new ring without s
  list RL = ringlist(A);
  RL = RL[1..4];
  list Lt = RL[2];
  Lt = delete(Lt,2*n+1);
  RL[2] = Lt;
  Lt = RL[3];
  Lt = delete(Lt,2);
  RL[3] = Lt;
  def @B = ring(RL);
  setring @B;
  def B = Weyl();
  setring B;
  kill @B;
  ideal LD = imap(A,LD);
  LD = engine(LD,eng);
  dbprint(ppl,"// ...done");
  dbprint(ppl-1,"//    Got: " + string(LD));
  kill A;
  intvec w = 1:n;
  ideal DR = deRhamCohomIdeal(LD,w,eng);
  export(DR);
  setring save;
  return(B);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y,z),dp;
  poly f = x^3+y^3+z^3;
  def A = deRhamCohom(f); // we see that the K-dim is 2
  setring A;
  DR;
}

// Appel hypergeometric functions /////////////////////////////////////////////

proc appelF1()
  "USAGE:  appelF1();
RETURN:  ring (a parametric Weyl algebra) containing an ideal 'IAppel1'
PURPOSE: defines the ideal in a parametric Weyl algebra,
@*       which annihilates Appel F1 hypergeometric function
NOTE:    The output ring is a parametric Weyl algebra. It contains an ideal
@*       'IAappel1' annihilating Appel F1 hypergeometric function.
@*       See (SST) p. 48.
EXAMPLE: example appelF1; shows example
"
{
  // 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);
  def @S = Weyl();
  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 (a parametric Weyl algebra) containing an ideal 'IAppel2'
PURPOSE: defines the ideal in a parametric Weyl algebra,
@*       which annihilates Appel F2 hypergeometric function
NOTE:    The output ring is a parametric Weyl algebra. It contains an ideal
@*       'IAappel2' annihilating Appel F2 hypergeometric function.
@*       See (SST) p. 85.
EXAMPLE: example appelF2; shows example
"
{
  // 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);
  def @S = Weyl();
  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 (a parametric Weyl algebra) containing an ideal 'IAppel4'
PURPOSE: defines the ideal in a parametric Weyl algebra,
@*       which annihilates Appel F4 hypergeometric function
NOTE:    The output ring is a parametric Weyl algebra. It contains an ideal
@*       'IAappel4' annihilating Appel F4 hypergeometric function.
@*       See (SST) p. 39.
EXAMPLE: example appelF4; shows example
"
{
  // 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);
  def @S = Weyl();
  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;
}


// characteric variety ////////////////////////////////////////////////////////

proc charVariety(ideal I, list #)
  "USAGE:  charVariety(I [,eng]); I an ideal, eng an optional int
RETURN:  ring (commutative) containing an ideal 'charVar'
PURPOSE: computes an ideal whose zero set is the characteristic variety of I in
@*       the sense of D-module theory
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
NOTE:    The output ring is commutative. It contains an ideal 'charVar'.
@*       If eng<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, and by default, @code{slimgb} is used.
DISPLAY: If @code{printlevel}=1, progress debug messages will be printed,
@*       if @code{printlevel}>=2, all the debug messages will be printed.
SEE ALSO: charInfo
EXAMPLE: example charVariety; shows examples
"
{
  // assumption check is done in GBWeight
  int eng;
  if (size(#)>0)
  {
    if (intLike(#[1]))
    {
      eng = int(#[1]);
    }
  }
  int ppl = printlevel - voice + 2;
  def save = basering;
  int n = nvars(save)/2;
  intvec u = 0:n;
  intvec v = 1:n;
  dbprint(ppl,"// Computing Groebner basis wrt weights...");
  I = GBWeight(I,u,v,eng);
  dbprint(ppl,"// ...finished");
  dbprint(ppl-1,"//    Got: " + string(I));
  list RL = ringlist(save);
  RL = RL[1..4];
  def newR = ring(RL);
  setring newR;
  ideal charVar = imap(save,I);
  intvec uv = u,v;
  charVar = inForm(charVar,uv);
  charVar = groebner(charVar);
  export(charVar);
  setring save;
  return(newR);
}
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 annihilator of F
  def CA = charVariety(LD);
  setring CA; CA; // commutative ring
  charVar;
  dim(charVar);   // hence I is holonomic
}

proc charInfo(ideal I)
  "USAGE:  charInfo(I);  I an ideal
RETURN:  ring (commut.) containing ideals 'charVar','singLoc' and list 'primDec'
PURPOSE: computes characteristic variety of I (in the sense of D-module theory),
@*       its singular locus and primary decomposition
ASSUME:  The basering is the n-th Weyl algebra over a field of 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).
NOTE:    In the output ring, which is commutative:
@*       - the ideal 'charVar' is the characteristic variety char(I),
@*       - the ideal 'SingLoc' is the singular locus of char(I),
@*       - the list 'primDec' is the primary decomposition of char(I).
DISPLAY: If @code{printlevel}=1, progress debug messages will be printed,
@*       if @code{printlevel}>=2, all the debug messages will be printed.
EXAMPLE: example charInfo; shows examples
"
{
  int ppl = printlevel - voice + 2;
  def save = basering;
  dbprint(ppl,"// computing characteristic variety...");
  def A = charVariety(I);
  setring A;
  dbprint(ppl,"// ...done");
  dbprint(ppl-1,"//    Got: " + string(charVar));
  dbprint(ppl,"// computing singular locus...");
  ideal singLoc = slocus(charVar);
  singLoc = groebner(singLoc);
  dbprint(ppl,"// ...done");
  dbprint(ppl-1,"//    Got: " + string(singLoc));
  dbprint(ppl,"// computing primary decomposition...");
  list primDec = primdecGTZ(charVar);
  dbprint(ppl,"// ...done");
  //export(charVar,singLoc,primDec);
  export(singLoc,primDec);
  setring save;
  return(A);
}
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 annihilator of F
  def CA = charInfo(LD);
  setring CA; CA; // commutative ring
  charVar;        // characteristic variety
  singLoc;        // singular locus
  primDec;        // primary decomposition
}


// 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);
  }

  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);
  }

  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;
  }
*/