source: git/Singular/LIB/bfun.lib @ 40e810d

//////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Noncommutative";
info="
LIBRARY: bfun.lib     Algorithms for b-functions and Bernstein-Sato polynomial
AUTHORS: Daniel Andres, daniel.andres@math.rwth-aachen.de
@* Viktor Levandovskyy,      levandov@math.rwth-aachen.de

THEORY: Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R,
@*      one is interested in the global b-function (also known as Bernstein-Sato
@*      polynomial) b(s) in K[s], defined to be the non-zero monic polynomial of minimal
@*      degree, satisfying a functional identity L * F^{s+1} = b(s) F^s,
@*      for some operator L in D[s] (* stands for the action of differential operator)
@*      By D one denotes the n-th Weyl algebra
@*      K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>.
@*      One is interested in the following data:
@*      - Bernstein-Sato polynomial b(s) in K[s],
@*      - the list of its roots, which are known to be rational
@*      - the multiplicities of the roots.
@*
@*   There is a constructive definition of a b-function of a holonomic ideal I in D
@*   (that is, an ideal I in a Weyl algebra D, such that D/I is holonomic module)
@*   with respect to the given weight vector w: For a polynomial p in D, its initial
@*   form w.r.t. (-w,w) is defined as the sum of all terms of p which have
@*   maximal weighted total degree where the weight of x_i is -w_i and the weight
@*   of d_i is w_i. Let J be the initial ideal of I w.r.t. (-w,w), i.e. the
@*   K-vector space generated by all initial forms w.r.t (-w,w) of elements of I.
@*   Put s = w_1 x_1 d_1 + ... + w_n x_n d_n. Then the monic generator b_w(s) of
@*   the intersection of J with the PID K[s] is called the b-function of I w.r.t.  w.
@*   Unlike Bernstein-Sato polynomial, general b-function with respect to
@*   arbitrary weights need not have rational roots at all. However, b-function
@*  of a holonomic ideal is known to be non-zero as well.
@*
@*      References: [SST] Saito, Sturmfels, Takayama: Groebner Deformations of
@*      Hypergeometric Differential Equations (2000),
@*      Noro: An Efficient Modular Algorithm for Computing the Global b-function,
@*      (2002).


MAIN PROCEDURES:

bfct(f[,s,t,v]);            compute the BS polynomial of f with linear reductions
bfctSyz(f[,r,s,t,u,v]);  compute the BS polynomial of f with syzygy-solver
bfctAnn(f[,s]);           compute the BS polynomial of f via Ann f^s + f
bfctOneGB(f[,s,t]);     compute the BS polynomial of f by just one GB computation
bfctIdeal(I,w[,s,t]);     compute the b-function of ideal w.r.t. weight
pIntersect(f,I[,s]);      intersection of ideal with subalgebra K[f] for a polynomial f
pIntersectSyz(f,I[,p,s,t]); intersection of ideal with subalgebra K[f] with syz-solver
linReduce(f,I[,s]);         reduce a polynomial by linear reductions w.r.t. ideal
linReduceIdeal(I,[s,t]);  interreduce generators of ideal by linear reductions
linSyzSolve(I[,s]);         compute a linear dependency of elements of ideal I

AUXILIARY PROCEDURES:

allPositive(v);  checks whether all entries of an intvec are positive
scalarProd(v,w); computes the standard scalar product of two intvecs
vec2poly(v[,i]); constructs an univariate polynomial with given coefficients

SEE ALSO: dmod_lib, dmodapp_lib, dmodvar_lib, gmssing_lib

KEYWORDS: D-module; global Bernstein-Sato polynomial; Bernstein-Sato polynomial; b-function; 
graded Weyl algebra; initial ideal; initial form; principal intersection; linear interreduction;
initial ideal approach;
";


LIB "qhmoduli.lib"; // for Max
LIB "dmod.lib";     // for SannfsBFCT etc
LIB "dmodapp.lib";  // for initialIdealW etc
LIB "nctools.lib";  // for isWeyl etc
LIB "presolve.lib"; // for valvars

///////////////////////////////////////////////////////////////////////////////
// testing for consistency of the library:
proc testbfunlib ()
{
  // tests all procs for consistency
  "AUXILIARY PROCEDURES:";
  example allPositive;
  example scalarProd;
  example vec2poly;
  "MAIN PROCEDURES:";
  example bfct;
  example bfctSyz;
  example bfctAnn;
  example bfctOneGB;
  example bfctIdeal;
  example pIntersect;
  example pIntersectSyz;
  example linReduce;
  example linReduceIdeal;
  example linSyzSolve;
}

//--------------- auxiliary procedures ----------------------------------------

/*
static proc gradedWeyl (intvec u,intvec v)
"USAGE:  gradedWeyl(u,v); u,v intvecs
RETURN:  a ring, the associated graded ring of the basering w.r.t. u and v
PURPOSE: compute the associated graded ring of the basering w.r.t. u and v
ASSUME: basering is a Weyl algebra
EXAMPLE: example gradedWeyl; shows examples
NOTE:    u[i] is the weight of x(i), v[i] the weight of D(i).
@*       u+v has to be a non-negative intvec.
"
{
  // todo check assumption
  int i;
  def save = basering;
  int n = nvars(save)/2;
  if (nrows(u)<>n || nrows(v)<>n)
  {
    ERROR("weight vectors have wrong dimension");
  }
  intvec uv,gr;
  uv = u+v;
  for (i=1; i<=n; i++)
  {
    if (uv[i]>=0)
    {
      if (uv[i]==0)
      {
        gr[i] = 0;
      }
      else
      {
        gr[i] = 1;
      }
    }
    else
    {
      ERROR("the sum of the weight vectors has to be a non-negative intvec");
    }
  }
  list l = ringlist(save);
  list l2 = l[2];
  matrix l6 = l[6];
  for (i=1; i<=n; i++)
  {
    if (gr[i] == 1)
    {
       l2[n+i] = "xi("+string(i)+")";
       l6[i,n+i] = 0;
    }
  }
  l[2] = l2;
  l[6] = l6;
  def G = ring(l);
  return(G);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring @D = 0,(x,y,z,Dx,Dy,Dz),dp;
  def D = Weyl();
  setring D;
  intvec u = -1,-1,1; intvec v = 2,1,1;
  def G = gradedWeyl(u,v);
  setring G; G;
}
*/

static proc safeVarName (string s)
{
  string 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 allPositive (intvec v)
"USAGE:  allPositive(v);  v an intvec
RETURN:  int, 1 if all components of v are positive, or 0 otherwise
PURPOSE: check whether all components of an intvec are positive
EXAMPLE: example allPositive; shows an example
"
{
  int i;
  for (i=1; i<=size(v); i++)
  {
    if (v[i]<=0)
    {
      return(0);
      break;
    }
  }
  return(1);
}
example
{
  "EXAMPLE:"; echo = 2;
  intvec v = 1,2,3;
  allPositive(v);
  intvec w = 1,-2,3;
  allPositive(w);
}

static proc findFirst (list l, i)
"USAGE:  findFirst(l,i);  l a list, i an argument of any type
RETURN:  int, the position of the first appearance of i in l,
@*       or 0 if i is not a member of l
PURPOSE: check whether the second argument is a member of a list
EXAMPLE: example findFirst; shows an example
"
{
  int j;
  for (j=1; j<=size(l); j++)
  {
    if (l[j]==i)
    {
      return(j);
      break;
    }
  }
  return(0);
}
//   findFirst(list(1, 2, list(1)),2);       // seems to be a bit buggy,
//   findFirst(list(1, 2, list(1)),3);       // but works for the purposes
//   findFirst(list(1, 2, list(1)),list(1)); // of this library
//   findFirst(l,list(2));
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  list l = 1,2,3;
  findFirst(l,4);
  findFirst(l,2);
}

proc scalarProd (intvec v, intvec w)
"USAGE:  scalarProd(v,w);  v,w intvecs
RETURN:  int, the standard scalar product of v and w
PURPOSE: computes the scalar product of two intvecs
ASSUME:  the arguments are of the same size
EXAMPLE: example scalarProd; shows examples
"
{
  int i; int sp;
  if (size(v)!=size(w))
  {
    ERROR("non-matching dimensions");
  }
  else
  {
    for (i=1; i<=size(v);i++)
    {
      sp = sp + v[i]*w[i];
    }
  }
  return(sp);
}
example
{
  "EXAMPLE:"; echo = 2;
  intvec v = 1,2,3;
  intvec w = 4,5,6;
  scalarProd(v,w);
}

//-------------- main procedures -------------------------------------------------------
proc linReduceIdeal(ideal I, list #)
"USAGE:  linReduceIdeal(I [,s,t,u]); I an ideal, s,t,u optional ints
RETURN:  ideal or list, linear reductum (over field) of I by its elements
PURPOSE: reduces a list of polys only by linear reductions (no monomial
@*        multiplications)
NOTE:    If s<>0, a list consisting of the reduced ideal and the coefficient
@*       vectors of the used reductions given as module is returned.
@*       Otherwise (and by default), only the reduced ideal is returned.
@*       If t<>0 (and by default) all monomials are reduced (if possible),
@*       otherwise, only leading monomials are reduced.
@*       If u<>0 (and by default), the ideal is first sorted in increasing order.
@*       If u is set to 0 and the given ideal is not sorted in the way described,
@*       the result might not be as expected.
DISPLAY: If @code{printlevel}=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example linReduceIdeal; shows examples
"
{
  // #[1] = remembercoeffs
  // #[2] = redtail
  // #[3] = sortideal
  int ppl = printlevel - voice + 2;
  int remembercoeffs = 0; // default
  int redtail        = 1; // default
  int sortideal      = 1; // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      remembercoeffs = #[1];
    }
    if (size(#)>1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        redtail = #[2];
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="int" || typeof(#[3])=="number")
        {
          sortideal = #[3];
        }
      }
    }
  }
  int sI = ncols(I);
  int sZeros = sI - size(I);
  int i,j;
  ideal J,lmJ,ordJ;
  list lJ = sort(I);
  module M; // for the coefficients
  // step 1: prepare, e.g. sort I
  if (sortideal <> 0)
  {
    if (sZeros > 0) // I contains zeros
    {
      if (remembercoeffs <> 0)
      {
        j = 0;
        for (i=1; i<=sI; i++)
        {
          if (I[i] == 0)
          {
            j++;
            J[j] = 0;
            ordJ[j] = -1;
            M[j] = gen(i);
          }
          else
          {
            M[i+sZeros-j] = gen(lJ[2][i-j]+j);
          }
        }
      }
      else
      {
        for (i=1; i<=sZeros; i++)
        {
          J[i] = 0;
          ordJ[i] = -1;
        }
      }
      I = J,lJ[1];
    }
    else // I contains no zeros
    {
      I = lJ[1];
      if (remembercoeffs <> 0)
      {
        for (i=1; i<=size(lJ[1]); i++) { M[i+sZeros] = gen(lJ[2][i]); }
      }
    }
  }
  else // assume I is already sorted
  {
    if (remembercoeffs <> 0)
    {
      for (i=1; i<=ncols(I); i++) { M[i] = gen(i);  }
    }
  }
  dbprint(ppl-1,"// initially sorted ideal:", I);
  if (remembercoeffs <> 0) { dbprint(ppl-1,"// used permutations:", M); }
  // step 2: reduce leading monomials by linear reductions
  poly lm,c,redpoly,maxlmJ;
  J[sZeros+1] = I[sZeros+1];
  lm = I[sZeros+1];
  maxlmJ = leadmonom(lm);
  lmJ[sZeros+1] = maxlmJ;
  int ordlm,reduction;
  ordJ[sZeros+1] = ord(lm);
  vector v;
  int noRedPast;
  for (i=sZeros+2; i<=sI; i++)
  {
    redpoly = I[i];
    lm = leadmonom(redpoly);
    ordlm = ord(lm);
    if (remembercoeffs <> 0) { v = M[i]; }
    reduction = 1;
    while (reduction == 1) // while there was a reduction
    {
      noRedPast = i;
      reduction = 0;
      for (j=sZeros+1; j<noRedPast; j++)
      {
        if (lm == 0) { break; } // nothing more to reduce
        if (lm > maxlmJ) { break; } //lm is bigger than maximal monomial to reduce with
        if (ordlm == ordJ[j])
        {
          if (lm == lmJ[j])
          {
            dbprint(ppl-1,"// reducing " + string(redpoly));
            dbprint(ppl-1,"//  with " + string(J[j]));
            c = leadcoef(redpoly)/leadcoef(J[j]);
            redpoly = redpoly - c*J[j];
            dbprint(ppl-1,"//  to " + string(redpoly));
            lm = leadmonom(redpoly);
            ordlm = ord(lm);
            if (remembercoeffs <> 0) { M[i] = M[i] - c * M[j]; }
            noRedPast = j;
            reduction = 1;
          }
        }
      }
    }
    for (j=sZeros+1; j<i; j++)
    {
      if (redpoly < J[j]) { break; }
    }
    J = insertGenerator(J,redpoly,j);
    lmJ = insertGenerator(lmJ,lm,j);
    ordJ = insertGenerator(ordJ,poly(ordlm),j);
    if (remembercoeffs <> 0)
    {
      v = M[i];
      M = deleteGenerator(M,i);
      M = insertGenerator(M,v,j);
    }
    maxlmJ = lmJ[i];
  }
  // step 3: reduce tails by linear reductions as well
  if (redtail <> 0)
  {
    dbprint(ppl,"// finished reducing leading monomials");
    dbprint(ppl-1,string(J));
    if (remembercoeffs <> 0) { dbprint(ppl-1,"// used reductions:" + string(M)); }
    int k;
    for (i=sZeros+1; i<=sI; i++)
    {
      lm = lmJ[i];
      for (j=i+1; j<=sI; j++)
      {
        for (k=2; k<=size(J[j]); k++) // run over all terms in J[j]
        {
          if (ordJ[i] == ord(J[j][k]))
          {
            if (lm == normalize(J[j][k]))
            {
              c = leadcoef(J[j][k])/leadcoef(J[i]);
              dbprint(ppl-1,"// reducing " + string(J[j]));
              dbprint(ppl-1,"//  with " + string(J[i]));
              J[j] = J[j] - c*J[i];
              dbprint(ppl-1,"//  to " + string(J[j]));
              if (remembercoeffs <> 0) { M[j] = M[j] - c * M[i]; }
            }
          }
        }
      }
    }
  }
  if (remembercoeffs <> 0) { return(list(J,M)); }
  else { return(J); }
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  ideal I = 3,x+9,y4+5x,2y4+7x+2;
  linReduceIdeal(I);     // reduces tails
  linReduceIdeal(I,0,0); // no reductions of tails
  list l = linReduceIdeal(I,1); // reduces tails and shows reductions used
  l;
  module M = I;
  matrix(l[1]) - M*l[2];
}

proc linReduce(poly f, ideal I, list #)
"USAGE:  linReduce(f, I [,s,t,u]); f a poly, I an ideal, s,t,u optional ints
RETURN:  poly or list, linear reductum (over field) of f by elements from I
PURPOSE: reduce a polynomial only by linear reductions (no monomial multiplications)
NOTE:    If s<>0, a list consisting of the reduced polynomial and the coefficient
@*       vector of the used reductions is returned, otherwise (and by default)
@*       only reduced polynomial is returned.
@*       If t<>0 (and by default) all monomials are reduced (if possible),
@*       otherwise, only leading monomials are reduced.
@*       If u<>0 (and by default), the ideal is linearly pre-reduced, i.e.
@*       instead of the given ideal, the output of @code{linReduceIdeal} is used.
@*       If u is set to 0 and the given ideal does not equal the output of
@*       @code{linReduceIdeal}, the result might not be as expected.
DISPLAY: If @code{printlevel}=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example linReduce; shows examples
"
{
  int ppl = printlevel - voice + 2;
  int remembercoeffs = 0; // default
  int redtail        = 1; // default
  int prepareideal   = 1; // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      remembercoeffs = #[1];
    }
    if (size(#)>1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        redtail = #[2];
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="int" || typeof(#[3])=="number")
        {
          prepareideal = #[3];
        }
      }
    }
  }
  int i,j,k;
  int sI = ncols(I);
  // pre-reduce I:
  module M;
  if (prepareideal <> 0)
  {
    if (remembercoeffs <> 0)
    {
      list sortedI = linReduceIdeal(I,1,redtail);
      I = sortedI[1];
      M = sortedI[2];
      dbprint(ppl-1,"// prepared ideal:" +string(I));
      dbprint(ppl-1,"//  with operations:" +string(M));
    }
    else { I = linReduceIdeal(I,0,redtail); }
  }
  else
  {
    if (remembercoeffs <> 0)
    {
      for (i=1; i<=sI; i++) { M[i] = gen(i); }
    }
  }
  ideal lmI,lcI,ordI;
  for (i=1; i<=sI; i++)
  {
    lmI[i] = leadmonom(I[i]);
    lcI[i] = leadcoef(I[i]);
    ordI[i] = ord(lmI[i]);
  }
  vector v;
  poly c;
  // === reduce leading monomials ===
  poly lm = leadmonom(f);
  int ordf = ord(lm);
  for (i=sI; i>=1; i--)  // I is sorted: smallest lm's on top
  {
    if (lm == 0) { break; }
    else
    {
      if (ordf == ordI[i])
      {
        if (lm == lmI[i])
        {
          c = leadcoef(f)/lcI[i];
          f = f - c*I[i];
          lm = leadmonom(f);
          ordf = ord(lm);
          if (remembercoeffs <> 0) { v = v - c * M[i]; }
        }
      }
    }
  }
  // === reduce tails as well ===
  if (redtail <> 0)
  {
    dbprint(ppl,"// finished reducing leading monomials");
    dbprint(ppl-1,string(f));
    if (remembercoeffs <> 0) { dbprint(ppl-1,"// used reductions:" + string(v)); }
    for (i=1; i<=sI; i++)
    {
      dbprint(ppl,"// testing ideal entry "+string(i));
      for (j=1; j<=size(f); j++)
      {
        if (ord(f[j]) == ordI[i])
        {
          if (normalize(f[j]) == lmI[i])
          {
            c = leadcoef(f[j])/lcI[i];
            f = f - c*I[i];
            dbprint(ppl-1,"// reducing with " + string(I[i]));
            dbprint(ppl-1,"//  to " + string(f));
            if (remembercoeffs <> 0) { v = v - c * M[i]; }
          }
        }
      }
    }
  }
  if (remembercoeffs <> 0)
  {
    list l = f,v;
    return(l);
  }
  else { return(f); }
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  ideal I = 1,y,xy;
  poly f = 5xy+7y+3;
  poly g = 7x+5y+3;
  linReduce(g,I);     // reduces tails
  linReduce(g,I,0,0); // no reductions of tails
  linReduce(f,I,1);   // reduces tails and shows reductions used
  f = x3+y2+x2+y+x;
  I = x3-y3, y3-x2,x3-y2,x2-y,y2-x;
  list l = linReduce(f,I,1);
  l;
  module M = I;
  f - (l[1]-(M*l[2])[1,1]);
}

proc linSyzSolve (ideal I, list #)
"USAGE:  linSyzSolve(I[,s]);  I an ideal, s an optional int
RETURN:  vector, coefficient vector of linear combination of 0 in elements of I
PURPOSE: compute a linear dependency between the elements of an ideal
@*       if such one exists
NOTE:    If s<>0, @code{std} is used for Groebner basis computations,
@*       otherwise, @code{slimgb} is used.
@*       By default, @code{slimgb} is used in char 0 and @code{std} in char >0.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example linSyzSolve; shows examples
"
{
  int whichengine     = 0; // default
  int enginespecified = 0; // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      whichengine = int( #[1]);
      enginespecified = 1;
    }
  }
  int ppl = printlevel - voice +2;
  int sI = ncols(I);
  // check if we are done
  if (I[sI]==0)
  {
    vector v = gen(sI);
  }
  else
  {
    // ------- 1. introduce undefined coeffs ------------------
    def save = basering;
    list RL = ringlist(save);
    int nv = nvars(save);
    int np = npars(save);
    int p = char(save);
    string cs = "(" + charstr(save) + ")";
    if (enginespecified == 0)
    {
      if (p <> 0)
      {
        whichengine = 1;
      }
    }
    int i;
    list Lvar;
    for (i=1; i<=sI; i++)
    {
      Lvar[i] = safeVarName("@a(" + string(i) + ")");
    }
    list L@A = RL[1..4];
    L@A[2] = Lvar;
    L@A[3] = list(list("lp",intvec(1:sI)),list("C",intvec(0)));
    def @A = ring(L@A);
    L@A[2] = list(safeVarName("@z"));
    L@A[3][1] = list("dp",intvec(1));
    if (size(L@A[1])>1)
    {
      L@A[1][2] = L@A[1][2] + Lvar;
      L@A[1][3][size(L@A[1][3])+1] = list("lp",intvec(1:sI));
    }
    else
    {
      L@A[1] = list(p,Lvar,list(list("lp",intvec(1:sI))),ideal(0));
    }
    def @aA = ring(L@A);
    def @B = save + @aA;
    setring @B;
    ideal I = imap(save,I);
    // ------- 2. form the linear system for the undef coeffs ---
     poly W;  ideal QQ;
    for (i=1; i<=sI; i++)
    {
      W = W + par(np+i)*I[i];
    }
    while (W!=0)
    {
      QQ = QQ,leadcoef(W);
      W = W - lead(W);
    }
    // QQ consists of polynomial expressions in @a(i) of type number
    setring @A;
    ideal QQ = imap(@B,QQ);
    // ------- 3. this QQ is a polynomial ideal, so "solve" the system -----
    dbprint(ppl, "// linSyzSolve: starting Groebner basis computation with engine:", whichengine);
    QQ = engine(QQ,whichengine);
    dbprint(ppl, "// QQ after engine:", QQ);
    if (dim(QQ) == -1)
    {
      dbprint(ppl+1, "// no solutions by linSyzSolve");
      // output zeroes
      setring save;
      kill @A,@aA,@B;
      return(v);
    }
    // ------- 4. in order to get the numeric values -------
    matrix AA = matrix(maxideal(1));
    module MQQ = std(module(QQ));
    AA = NF(AA,MQQ); // todo: we still receive NF warnings
    dbprint(ppl, "// AA after NF:",AA);
    //    "AA after NF:"; print(AA);
    for(i=1; i<=sI; i++)
    {
      AA = subst(AA,var(i),1);
    }
    dbprint(ppl, "// AA after subst:",AA);
    //    "AA after subst: "; print(AA);
    vector v = (module(transpose(AA)))[1];
    setring save;
    vector v = imap(@A,v);
    kill @A,@aA,@B;
  }
  return(v);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,x,dp;
  ideal I = x,2x;
  linSyzSolve(I);
  ideal J = x,x2;
  linSyzSolve(J);
}

proc pIntersect (poly s, ideal I, list #)
"USAGE:  pIntersect(f, I [,s]);  f a poly, I an ideal, s an optional int
RETURN:  vector, coefficient vector of the monic polynomial
PURPOSE: compute the intersection of ideal I with the subalgebra K[f]
ASSUME:  I is given as Groebner basis, basering is not a qring.
NOTE:    If the intersection is zero, this proc might not terminate.
@*       If s>0 is given, it is searched for the generator of the intersection
@*       only up to degree s. Otherwise (and by default), no bound is assumed.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example pIntersect; shows examples
"
{
  def save = basering;
  int n = nvars(save);
  list RL = ringlist(save);
  int i,j,k;
  if (RL[4]<>0)
  {
    ERROR ("basering must not be a qring");
  }
  // assume I is given in Groebner basis
  if (attrib(I,"isSB") <> 1)
  {
    print("// WARNING: The input has no SB attribute!");
    print("// Treating it as if it were a Groebner basis and proceeding...");
    attrib(I,"isSB",1); // set attribute for suppressing NF messages
  }
  int bound = 0; // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      bound = #[1];
    }
  }
  int ppl = printlevel-voice+2;
  // ---case 1: I = basering---
  if (size(I) == 1)
  {
    if (simplify(I,2)[1] == 1)
    {
      return(gen(1)); // = 1
    }
  }
  // ---case 2: intersection is zero---
  intvec degs = leadexp(s);
  intvec possdegbounds;
  list degI;
  i = 1;
  while (i <= ncols(I))
  {
    if (i == ncols(I)+1) { break; }
    degI[i] = leadexp(I[i]);
    for (j=1; j<=n; j++)
    {
      if (degs[j] == 0)
      {
        if (degI[i][j] <> 0) { break; }
      }
      if (j == n)
      {
        k++;
        possdegbounds[k] = Max(degI[i]);
      }
    }
    i++;
  }
  int degbound = Min(possdegbounds);
  if (bound>0 && bound<degbound) // given bound is too small
  {
    print("// Try a bound of at least " + string(degbound));
    return(vector(0));
  }
  dbprint(ppl,"// lower bound for the degree of the insection is " +string(degbound));
  if (degbound == 0) // lm(s) does not appear in lm(I)
  {
    print("// Intersection is zero");
    return(vector(0));
  }
  // ---case 3: intersection is non-trivial---
  ideal redNI = 1;
  vector v;
  list l,ll;
  l[1] = vector(0);
  poly toNF,tobracket,newNF,rednewNF,oldNF,secNF;
  i = 1;
  while (1)
  {
    if (bound>0 && i>bound) { return(vector(0)); }
    dbprint(ppl,"// Testing degree "+string(i));
    if (i>1)
    {
      oldNF = newNF;
      tobracket = s^(i-1) - oldNF;
      if (tobracket==0) // todo bug in bracket?
      {
        toNF = 0;
      }
      else
      {
        toNF = bracket(tobracket,secNF);
      }
      newNF = NF(toNF+oldNF*secNF,I);  // = NF(s^i,I)
    }
    else
    {
      newNF = NF(s,I);
      secNF = newNF;
    }
    ll = linReduce(newNF,redNI,1);
    rednewNF = ll[1];
    l[i+1] = ll[2];
    dbprint(ppl-1,"// newNF is: "+string(newNF));
    dbprint(ppl-1,"// rednewNF is: "+string(rednewNF));
    if (rednewNF != 0) // no linear dependency
    {
      redNI[i+1] = rednewNF;
      i++;
    }
    else // there is a linear dependency, hence we are done
    {
      dbprint(ppl,"// degree of the generator of the intersection is: "+string(i));
      break;
    }
  }
  dbprint(ppl-1,"// used linear reductions:", l);
  // we obtain the coefficients of the generator by the used reductions:
  list Lvar;
  for (j=1; j<=i+1; j++)
  {
    Lvar[j] = safeVarName("a("+string(j)+")");
  }
  list Lord = list("dp",intvec(1:(i+1))),list("C",intvec(0));
  list L@R = RL[1..4];
  L@R[2] = Lvar; L@R[3] = Lord;
  def @R = ring(L@R); setring @R;
  list l = imap(save,l);
  ideal C;
  for (j=1;j<=i+1;j++)
  {
    C[j] = 0;
    for (k=1;k<=j;k++)
    {
      C[j] = C[j]+l[j][k]*var(k);
    }
  }
  for (j=i;j>=1;j--)
  {
    C[i+1]= subst(C[i+1],var(j),var(j)+C[j]);
  }
  matrix m = coeffs(C[i+1],maxideal(1));
  vector v = gen(i+1);
  for (j=1;j<=i+1;j++)
  {
    v = v + m[j,1]*gen(j);
  }
  setring save;
  v = imap(@R,v);
  kill @R;
  return(v);
}
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;
  pIntersect(t*Dt,inF);
  pIntersect(t*Dt,inF,1);
}

proc pIntersectSyz (poly s, ideal I, list #)
"USAGE:  pIntersectSyz(f, I [,p,s,t]);  f poly, I ideal, p,t optial ints, p prime
RETURN:  vector, coefficient vector of the monic polynomial
PURPOSE: compute the intersection of an ideal I with the subalgebra K[f]
ASSUME:  I is given as Groebner basis.
NOTE:    If the intersection is zero, this procedure might not terminate.
@*       If p>0 is given, this proc computes the generator of the intersection in
@*       char p first and then only searches for a generator of the obtained
@*       degree in the basering. Otherwise, it searches for all degrees by
@*       computing syzygies.
@*       If s<>0, @code{std} is used for Groebner basis computations in char 0,
@*       otherwise, and by default, @code{slimgb} is used.
@*       If t<>0 and by default, @code{std} is used for Groebner basis
@*       computations in char >0, otherwise, @code{slimgb} is used.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example pIntersectSyz; shows examples
"
{
  // assume I is given as Groebner basis
  if (attrib(I,"isSB") <> 1)
  {
    print("// WARNING: The input has no SB attribute!");
    print("//  Treating it as if it were a Groebner basis and proceeding...");
    attrib(I,"isSB",1); // set attribute for suppressing NF messages
  }
  int ppl = printlevel-voice+2;
  int whichengine  = 0; // default
  int modengine    = 1; // default
  int solveincharp = 0; // default
  def save = basering;
  int n = nvars(save);
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      solveincharp = int(#[1]);
    }
    if (size(#)>1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        whichengine = int(#[2]);
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="int" || typeof(#[3])=="number")
        {
          modengine = int(#[3]);
        }
      }
    }
  }
  int i,j;
  vector v;
  poly tobracket,toNF,newNF,p;
  ideal NI = 1;
  newNF = NF(s,I);
  NI[2] = newNF;
  list RL = ringlist(save);
  if (solveincharp)
  {
    int psolveincharp = prime(solveincharp);
    if (solveincharp <> psolveincharp)
    {
      print("// " + string(solveincharp) + " is invalid characteristic of ground field.");
      print("// " + string(psolveincharp) + " is used.");
      solveincharp = psolveincharp;
      kill psolveincharp;
    }
    list RLp = RL[1..4];
    if (typeof(RL[1]) == "int") { RLp[1] = solveincharp; }
    else { RLp[1][1] = solveincharp; } // parameters
    def @Rp = ring(RLp);
    setring @Rp;
    number c;
    setring save;
    int shortSave = short; // workaround for maps Q(a_i) -> Z/p(a_i)
    short = 0;
    string str;
    int badprime;
    i=1;
    while (badprime == 0 && i<=size(s)) // detect bad primes
    {
      str = string(denominator(leadcoef(s[i])));
      str = "c = " + str + ";";
      setring @Rp;
      execute(str);
      if (c == 0) { badprime = 1; }
      setring save;
      i++;
    }
    str = "poly s = " + string(s) + ";";
    if (size(RL) > 4) // basering is NC-algebra
    {
      string RL5 = "@C = " + string(RL[5]) + ";";
      string RL6 = "@D = " + string(RL[6]) + ";";
      setring @Rp;
      matrix @C[n][n]; matrix @D[n][n];
      execute(RL5); execute(RL6);
      def Rp = nc_algebra(@C,@D);
    }
    else { def Rp = @Rp; }
    setring Rp;
    kill @Rp;
    dbprint(ppl-1,"// solving in ring ", Rp);
    execute(str);
    vector v;
    number c;
    ideal NI = 1;
    setring save;
    i=1;
    while (badprime == 0 && i<=size(I)) // detect bad primes
    {
      str = string(leadcoef(cleardenom(I[i])));
      str = "c = " + str + ";";
      setring Rp;
      execute(str);
      if (c == 0) { badprime = 1; }
      setring save;
      i++;
    }
    if (badprime == 1)
    {
      print("// WARNING: bad prime");
      short = shortSave;
      return(vector(0));
    }
  }
  i = 1;
  dbprint(ppl,"// pIntersectSyz starts...");
  dbprint(ppl-1,"// with ideal I=", I);
  while (1)
  {
    dbprint(ppl,"// testing degree: "+string(i));
    if (i>1)
    {
      tobracket = s^(i-1)-NI[i];
      if (tobracket!=0)
      {
        toNF = bracket(tobracket,NI[2]) + NI[i]*NI[2];
      }
      else
      {
        toNF = NI[i]*NI[2];
      }
      newNF =  NF(toNF,I);
      NI[i+1] = newNF;
    }
    // look for a solution
    dbprint(ppl-1,"// linSyzSolve starts with: "+string(matrix(NI)));
    if (solveincharp) // modular method
    {
      for (j=1; j<=size(newNF); j++)
      {
        str = string(denominator(leadcoef(s[i])));
        str = "c = " + str + ";";
        setring Rp;
        execute(str);
        if (c == 0)
        {
          print("// WARNING: bad prime");
          setring save;
          short = shortSave;
          return(vector(0));
        }
        setring save;
      }
      str = "NI[" + string(i) + "+1] = " + string(newNF) + ";";
      setring Rp;
      execute(str); // NI[i+1] = [newNF]_{solveincharp}
      v = linSyzSolve(NI,modengine);
      if (v!=0) // there is a modular solution
      {
        dbprint(ppl,"// got solution in char "+string(solveincharp)+" of degree "+string(i));
        setring save;
        v = linSyzSolve(NI,whichengine);
        if (v==0)
        {
          break;
        }
      }
      else // no modular solution
      {
        setring save;
        v = 0;
      }
    }
    else // non-modular method
    {
      v = linSyzSolve(NI,whichengine);
    }
    matrix MM[1][nrows(v)] = matrix(v);
    dbprint(ppl-1,"// linSyzSolve ready  with: "+string(MM));
    kill MM;
    //  "linSyzSolve ready with"; print(v);
    if (v!=0)
    {
      // a solution:
      //check for the reality of the solution
      p = 0;
      for (j=1; j<=i+1; j++)
      {
        p = p + v[j]*NI[j];
      }
      if (p!=0)
      {
        dbprint(ppl,"// linSyzSolve: bad solution!");
      }
      else
      {
        dbprint(ppl,"// linSyzSolve: got solution!");
        // "got solution!";
        break;
      }
    }
    // no solution:
    i++;
  }
  dbprint(ppl,"// pIntersectSyz finished");
  if (solveincharp) { short = shortSave; }
  return(v);
}
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;
  poly s = t*Dt;
  pIntersectSyz(s,inF);
  int p = prime(20000);
  pIntersectSyz(s,inF,p,0,0);
}

proc vec2poly (list #)
"USAGE:  vec2poly(v [,i]);  v a vector or an intvec, i an optional int
RETURN:  poly, an univariate polynomial in i-th variable with coefficients given by v
PURPOSE: constructs an univariate polynomial in K[var(i)] with given coefficients,
@*       such that the coefficient at var(i)^{j-1} is v[j].
NOTE:    The optional argument i must be positive, by default i is 1.
EXAMPLE: example vec2poly; shows examples
"
{
  def save = basering;
  int i,ringvar;
  ringvar = 1; // default
  if (size(#) > 0)
  {
    if (typeof(#[1])=="vector" || typeof(#[1])=="intvec")
    {
      def v = #[1];
    }
    else
    {
      ERROR("wrong input: expected vector/intvec expression");
    }
    if (size(#) > 1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        ringvar = int(#[2]);
      }
    }
  }
  if (ringvar > nvars(save))
  {
    ERROR("var out of range");
  }
  poly p;
  for (i=1; i<=nrows(v); i++)
  {
    p = p + v[i]*(var(ringvar))^(i-1);
  }
  return(p);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  vector v = gen(1) + 3*gen(3) + 22/9*gen(4);
  intvec iv = 3,2,1;
  vec2poly(v,2);
  vec2poly(iv);
}

static proc listofroots (list #)
{
  def save = basering;
  int n = nvars(save);
  int i;
  poly p;
  if (typeof(#[1])=="vector")
  {
    vector b = #[1];
    for (i=1; i<=nrows(b); i++)
    {
      p = p + b[i]*(var(1))^(i-1);
    }
  }
  else
  {
    p = #[1];
  }
  int substitution = int(#[2]);
  string s = safeVarName("s");
  list RL = ringlist(save); RL = RL[1..4];
  RL[2] = list(s); RL[3] = list(list("dp",intvec(1)),list("C",0));
  def S = ring(RL); setring S;
  ideal J;
  for (i=1; i<=n; i++)
  {
    J[i] = var(1);
  }
  map @m = save,J;
  poly p = @m(p);
  if (substitution == 1)
  {
    p = subst(p,var(1),-var(1)-1);
  }
  // the rest of this proc is nicked from bernsteinBM from dmod.lib
  list P = factorize(p);//with constants and multiplicities
  ideal bs; intvec m;   //the BS polynomial is monic, so we are not interested in constants
  for (i=2; i<= size(P[1]); i++)  //we delete P[1][1] and P[2][1]
  {
    bs[i-1] = P[1][i];
    m[i-1]  = P[2][i];
  }
  bs =  normalize(bs);
  bs = -subst(bs,var(1),0);
  setring save;
  ideal bs = imap(S,bs);
  kill S;
  list BS = bs,m;
  return(BS);
}

static proc addRoot(number q, list L)
{
  // add root to list in bFactor format
  int i,qInL;
  ideal I = L[1];
  intvec v = L[2];
  list LL;
  if (v == 0)
  {
    I = poly(q);
    v = 1;
    LL = I,v;
  }
  else
  {
    for (i=1; i<=ncols(I); i++)
    {
      if (I[i] == q)
      {
        qInL = i;
        break;
      }
    }
    if (qInL)
    {
      v[qInL] = v[qInL] + 1;
    }
    else
    {
      I = q,I;
      v = 1,v;
    }
  }
  LL = I,v;
  if (size(L) == 3) // irreducible factor
  {
    if (L[3] <> "0" && L[3] <> "1")
    {
      LL = LL + list(L[3]);
    }
  }
  return(LL);
}

static proc bfctengine (poly f, int inorann, int whichengine, int addPD, int stdsum, int methodord, int methodpIntersect, int pIntersectchar, int modengine, intvec u0)
{
  int ppl = printlevel - voice +2;
  int i;
  def save = basering;
  int n = nvars(save);
  if (isCommutative() == 0) { ERROR("basering must be commutative"); }
  if (char(save) <> 0) { ERROR("characteristic of basering has to be 0"); }
  list L = ringlist(save);
  int qr;
  if (L[4] <> 0) // qring
  {
    print("// basering is qring:");
    print("// discarding the quotient and proceeding...");
    L[4] = ideal(0);
    qr = 1;
    def save2 = ring(L); setring save2;
    poly f = imap(save,f);
  }
  if (inorann == 0) // bfct using initial ideal
  {
    // list L = ringlist(basering);
    intvec iv = valvars(f)[1]; // heuristacally better ordering for initialMalgrange
    list varL = L[2];
    varL = varL[iv];
    L[2] = varL;
    if (u0 <> 0)
    {
      u0 = u0[iv];
    }
    def newr = ring(L);
    kill varL,iv,L;
    setring newr;
    poly f = imap(save,f);
    def D = initialMalgrange(f,whichengine,methodord,u0);
    setring D;
    ideal J = inF;
    kill inF;
    poly s = t*Dt;
  }
  else // bfct using Ann(f^s)
  {
    def D = SannfsBFCT(f,addPD,whichengine,stdsum);
    setring D;
    ideal J = LD;
    kill LD;
    poly s = var(1);
  }
  vector b;
  // try it modular
  if (methodpIntersect <> 0) // pIntersectSyz
  {
    if (pIntersectchar == 0) // pIntersectSyz::modular
    {
      list L = ringlist(D);
      int lb = 10000; int ub = 536870909; // bounds for random primes
      list usedprimes;
      int sJ = size(J);
      int sLJq;
      ideal LJ;
      for (i=1; i<=sJ; i++)
      {
        LJ[i] = leadcoef(cleardenom(J[i]));
      }
      int short_save = short; // workaround for map Q(a_i) -> Z/q(a_i)
      short = 0;
      string strLJq = "ideal LJq = " + string(LJ) + ";";
      int nD = nvars(D);
      string L5 = "matrix @C[nD][nD]; @C = " + string(L[5]) + ";";
      string L6 = "matrix @D[nD][nD]; @D = " + string(L[6]) + ";";
      L = L[1..4];
      i = 1;
      while (b == 0)
      {
        dbprint(ppl,"// number of run in the loop: "+string(i));
        int q = prime(random(lb,ub));
        if (findFirst(usedprimes,q)==0) // if q has not been used already
        {
          usedprimes = usedprimes,q;
          dbprint(ppl,"// using prime: "+string(q));
          if (typeof(L[1]) == "int") { L[1] = q; }
          else { L[1][1] = q; } // parameters
          def @Rq = ring(L); setring @Rq;
          execute(L5); execute(L6);
          def Rq = nc_algebra(@C,@D); // def Rq = nc_algebra(1,@D);
          setring Rq; kill @Rq;
          execute(strLJq);
          sLJq = size(LJq);
          setring D; kill Rq;
          if (sLJq <> sJ ) // detect unlucky prime
          {
            dbprint(ppl,"// " +string(q) + " is unlucky");
            b = 0;
          }
          else
          {
            b = pIntersectSyz(s,J,q,whichengine,modengine);
          }
        }
        i++;
      }
      short = short_save;
    }
    else // pIntersectSyz::non-modular
    {
      b = pIntersectSyz(s,J,0,whichengine);
    }
  }
  else // pIntersect: linReduce
  {
    b = pIntersect(s,J);
  }
  if (inorann == 0) { list l = listofroots(b,1); }
  else // bfctAnn
  {
    list l = listofroots(b,0);
    if (addPD)
    {
      l = addRoot(-1,l);
    }
  }
  setring save;
  list l = imap(D,l);
  return(l);
}

proc bfct (poly f, list #)
"USAGE:  bfct(f [,s,t,v]);  f a poly, s,t optional ints, v an optional intvec
RETURN:  list of ideal and intvec
PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s)
@*       for the hypersurface defined by f.
ASSUME:  The basering is commutative and of characteristic 0.
BACKGROUND: In this proc, the initial Malgrange ideal is computed according to
@*       the algorithm by Masayuki Noro and then a system of linear equations is
@*       solved by linear reductions.
NOTE:    In the output list, the ideal contains all the roots
@*       and the intvec their multiplicities.
@*       If s<>0, @code{std} is used for GB 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 v is a positive weight vector, v is used for homogenization
@*       computations, 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 bfct; shows examples
"
{
  int ppl = printlevel - voice +2;
  int i;
  int n = nvars(basering);
  // in # we have two switches:
  // one for the engine used for Groebner basis computations,
  // one for  M() ordering or its realization
  // in # can also be the optional weight vector
  int whichengine  = 0; // default
  int methodord    = 0; // default
  intvec u0        = 0; // 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];
        }
      }
    }
  }
  list b = bfctengine(f,0,whichengine,0,0,methodord,0,0,0,u0);
  return(b);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly f = x^2+y^3+x*y^2;
  bfct(f);
  intvec v = 3,2;
  bfct(f,1,0,v);
}

proc bfctSyz (poly f, list #)
"USAGE:  bfctSyz(f [,r,s,t,u,v]);  f poly, r,s,t,u optional ints, v opt. intvec
RETURN:  list of ideal and intvec
PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s)
@*       for the hypersurface defined by f
ASSUME:  The basering is commutative and of characteristic 0.
BACKGROUND: In this proc, the initial Malgrange ideal is computed according to
@*       the algorithm by Masayuki Noro and then a system of linear equations is
@*       solved by computing syzygies.
NOTE:    In the output list, the ideal contains all the roots and the intvec
@*       their multiplicities.
@*       If r<>0, @code{std} is used for GB computations in characteristic 0,
@*       otherwise, and by default, @code{slimgb} is used.
@*       If s<>0, a matrix ordering is used for GB computations, otherwise,
@*       and by default, a block ordering is used.
@*       If t<>0, the computation of the intersection is solely performed over
@*       charasteristic 0, otherwise and by default, a modular method is used.
@*       If u<>0 and by default, @code{std} is used for GB computations in
@*       characteristic >0, otherwise, @code{slimgb} is used.
@*       If v is a positive weight vector, v is used for homogenization
@*       computations, 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 bfctSyz; shows examples
"
{
  int ppl = printlevel - voice +2;
  int i;
  // in # we have four switches:
  // one for the engine used for Groebner basis computations in char 0,
  // one for  M() ordering or its realization
  // one for a modular method when computing the intersection
  // and one for the engine used for Groebner basis computations in char >0
  // in # can also be the optional weight vector
  int n = nvars(basering);
  int whichengine = 0; // default
  int methodord   = 0; // default
  int pIntersectchar  = 0; // default
  int modengine   = 1; // default
  intvec u0       = 0; // 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])=="int" || typeof(#[3])=="number")
        {
          pIntersectchar = int(#[3]);
        }
        if (size(#)>3)
        {
          if (typeof(#[4])=="int" || typeof(#[4])=="number")
          {
            modengine = int(#[4]);
          }
          if (size(#)>4)
          {
            if (typeof(#[5])=="intvec" && size(#[5])==n && allPositive(#[5])==1)
            {
              u0 = #[5];
            }
          }
        }
      }
    }
  }
  list b = bfctengine(f,0,whichengine,0,0,methodord,1,pIntersectchar,modengine,u0);
  return(b);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly f = x^2+y^3+x*y^2;
  bfctSyz(f);
  intvec v = 3,2;
  bfctSyz(f,0,1,1,0,v);
}

proc bfctIdeal (ideal I, intvec w, list #)
"USAGE:  bfctIdeal(I,w[,s,t]);  I an ideal, w an intvec, s,t optional ints
RETURN:  list of ideal and intvec
PURPOSE: computes the roots of the global b-function of I w.r.t. the weight (-w,w).
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).
@*       Further we assume that I is holonomic.
BACKGROUND:  In this proc, the initial ideal of I is computed according to the
@*       algorithm by Masayuki Noro and then a system of linear equations is
@*       solved by linear reductions.
NOTE:    In the output list, say L,
@*  - L[1] of type ideal contains all the rational roots of a b-function,
@*  - L[2] of type intvec contains the multiplicities of above roots,
@*  - optional L[3] of type string is the part of b-function without rational roots.
@*  Note, that a b-function of degree 0 is encoded via L[1][1]=0, L[2]=0 and
@*  L[3] is 1 (for nonzero constant) or 0 (for zero b-function).
@*       If s<>0, @code{std} is used for GB computations in characteristic 0,
@*       otherwise, and by default, @code{slimgb} is used.
@*       If t<>0, a matrix ordering is used for GB computations, otherwise,
@*       and by default, a block ordering is used.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example bfctIdeal; shows examples
"
{
  int ppl = printlevel - voice +2;
  int i;
  def save = basering;
  int n = nvars(save)/2;
  int whichengine = 0; // default
  int methodord   = 0; // 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 (isWeyl()==0) { ERROR("basering is not a Weyl algebra"); }
  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)));
    }
  }
  int isH = isHolonomic(I);
  if (isH<>1)
  {
    print("WARNING: given ideal is not holonomic");
    print("... setting bound for degree of b-function to 10 and proceeding");
    isH = 10;
  }
  else { isH = 0; } // no degree bound for pIntersect
  ideal J = initialIdealW(I,-w,w,whichengine,methodord);
  poly s;
  for (i=1; i<=n; i++)
  {
    s = s + w[i]*var(i)*var(n+i);
  }
  vector b = pIntersect(s,J,isH);
  list RL = ringlist(save); RL = RL[1..4];
  RL[2] = list(safeVarName("s"));
  RL[3] = list(list("dp",intvec(1)),list("C",intvec(0)));
  def @S = ring(RL); setring @S;
  vector b = imap(save,b);
  poly bs = vec2poly(b);
  list l = bFactor(bs);
  setring save;
  list l = imap(@S,l);
  return(l);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring @D = 0,(x,y,Dx,Dy),dp;
  def D = Weyl();
  setring D;
  ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6; I = std(I);
  intvec w1 = 0,1;
  intvec w2 = 2,3;
  bfctIdeal(I,w1);
  bfctIdeal(I,w2,0,1);
  ideal J = I[size(I)]; // J is not holonomic by construction
  bfctIdeal(J,w1); //  b-function of D/J w.r.t. w1 is non-zero
  bfctIdeal(J,w2); //  b-function of D/J w.r.t. w2 is zero
}

proc bfctOneGB (poly f,list #)
"USAGE:  bfctOneGB(f [,s,t]);  f a poly, s,t optional ints
RETURN:  list of ideal and intvec
PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) for the
@*       hypersurface defined by f, using only one GB computation
ASSUME:  The basering is commutative and of characteristic 0.
BACKGROUND: In this proc, the initial Malgrange ideal is computed based on the
@*       algorithm by Masayuki Noro and combined with an elimination ordering.
NOTE:    In the output list, the ideal contains all the roots and the intvec
@*       their multiplicities.
@*       If s<>0, @code{std} is used for the GB computation, otherwise,
@*       and by default, @code{slimgb} is used.
@*       If t<>0, a matrix ordering is used for GB computations,
@*       otherwise, and by default, a block ordering is used.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example bfctOneGB; shows examples
"
{
  int ppl = printlevel - voice +2;
  if (!isCommutative()) { ERROR("Basering must be commutative"); }
  def save = basering;
  int n = nvars(save);
  if (char(save) <> 0)
  {
    ERROR("characteristic of basering has to be 0");
  }
  list L = ringlist(save);
  int qr;
  if (L[4] <> 0) // qring?
  {
    print("// basering is qring:");
    print("// discarding the quotient and proceeding...");
    L[4] = ideal(0);
    qr = 1;
    def save2 = ring(L);
    setring save2;
    poly f = imap(save,f);
  }
  int N = 2*n+4;
  int i;
  int whichengine = 0; // default
  int methodord   = 0; // 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]);
      }
    }
  }
  // creating the homogenized extended Weyl algebra
  // create names for vars
  list Lvar;
  Lvar[1]   = safeVarName("t");
  Lvar[2]   = safeVarName("s");
  Lvar[n+3] = safeVarName("D"+Lvar[1]);
  Lvar[N]   = safeVarName("h");
  for (i=1; i<=n; i++)
  {
    Lvar[i+2]   = string(var(i));
    Lvar[i+n+3] = safeVarName("D" + string(var(i)));
  }
  // create ordering
  intvec uv = -1; uv[n+3] = 1; uv[N] = 0;
  intvec @a = 1:N; @a[2] = 2;
  intvec @a2 = @a; @a2[2] = 0; @a2[2*n+4] = 0;
  list Lord;
  Lord[1] = list("a",@a); Lord[2] = list("a",@a2);
  if (methodord == 0) // default: block ordering
  {
    //ring @Dh = 0,(t,s,x(n..1),Dt,D(n..1),h),(a(@a),a(@a2),a(uv),dp(N-1),lp(1));
    Lord[3] = list("a",uv);
    Lord[4] = list("dp",intvec(1:(N-1)));
    Lord[5] = list("lp",intvec(1));
    Lord[6] = list("C",intvec(0));
  }
  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,"// weights for ordering: "+string(transpose(@a)));
    dbprint(ppl,"// the ordering matrix:",@Ord);
    //ring @Dh = 0,(t,s,x(n..1),Dt,D(n..1),h),(a(@a),a(@a2),M(@Ord));
    Lord[3] = list("M",intvec(@Ord));
    Lord[4] = list("C",intvec(0));
  }
  // create commutative ring
  list L@Dh = ringlist(basering);
  L@Dh = L@Dh[1..4]; // if basering is commutative nc_algebra
  L@Dh[2] = Lvar; L@Dh[3] = Lord;
  def @Dh = ring(L@Dh); setring @Dh;
  dbprint(ppl,"// the ring @Dh:",@Dh);
  // create non-commutative relations
  matrix @relD[N][N];
  @relD[1,2] = var(1)*var(N)^2;     //  s*t = t*s + t*h^2
  @relD[2,n+3] = var(n+3)*var(N)^2; // Dt*s = s*Dt+Dt*h^2
  @relD[1,n+3] = var(N)^2;
  for (i=1; i<=n; i++)
  {
    @relD[i+2,n+3+i] = var(N)^2;
  }
  dbprint(ppl,"// nc relations:",@relD);
  def Dh = nc_algebra(1,@relD);
  setring Dh; kill @Dh;
  dbprint(ppl,"// computing in ring",Dh);
  poly f = imap(save,f);
  f = homog(f,h);
  // create the Malgrange ideal
  ideal I = var(1) - f, var(1)*var(n+3) - var(2);
  for (i=1; i<=n; i++)
  {
    I[3+i] = var(i+n+3)+diff(f,var(i+2))*var(n+3);
  }
  dbprint(ppl-1, "// the Malgrange ideal: " +string(I));
  // 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");
  I = subst(I,h,1); // dehomogenization
  dbprint(ppl-1,string(I));
  // 3.3 the initial form
  I = inForm(I,uv);
  dbprint(ppl, "// the initial ideal:", string(matrix(I)));
  // read off the solution
  intvec tonselect = 1;
  for (i=3; i<=2*n+4; i++) { tonselect = tonselect,i; }
  I = nselect(I,tonselect);
  dbprint(ppl, "// generators containing only s:", string(matrix(I)));
  I = engine(I, whichengine); // is now a principal ideal;
  if (qr == 1) { setring save2; }
  else { setring save; }
  ideal J; J[2] = var(1);
  map @m = Dh,J;
  ideal I = @m(I);
  poly p = I[1];
  list l = listofroots(p,1);
  if (qr == 1)
  {
    setring save;
    list l = imap(save2,l);
  }
  return(l);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly f = x^2+y^3+x*y^2;
  bfctOneGB(f);
  bfctOneGB(f,1,1);
}



proc bfctAnn (poly f, list #)
"USAGE:  bfctAnn(f [,a,b,c]);  f a poly, a, b, c optional ints
RETURN:  list of ideal and intvec
PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) for the
@*       hypersurface defined by f.
ASSUME:  The basering is commutative and of characteristic 0.
BACKGROUND: In this proc, Ann(f^s) is computed and then a system of linear
@*       equations is solved by linear reductions.
NOTE:    In the output list, the ideal contains all the roots and the intvec
@*  their multiplicities.
@*  If a<>0, only f is appended to Ann(f^s), otherwise, and by default,
@*  f and all its partial derivatives are appended.
@*  If b<>0, @code{std} is used for GB computations, otherwise, and by
@*  default, @code{slimgb} is used.
@*  If c<>0, @code{std} is used for Groebner basis computations of ideals
@*  <I+J> when I is already a Groebner basis of <I>.
@*  Otherwise, and by default the engine determined by the switch b is used.
@*  Note that in the case c<>0, the choice for b will be overwritten only
@*  for the types of ideals mentioned above.
@*  This means that if b<>0, specifying c has no effect.
DISPLAY: If printlevel=1, progress debug messages will be printed,
@*       if printlevel>=2, all the debug messages will be printed.
EXAMPLE: example bfctAnn; shows examples
"
{
  def save = basering;
  int ppl = printlevel - voice + 2;
  int addPD       = 1; // default
  int whichengine = 0; // default
  int stdsum      = 0; // default
  if (size(#)>0)
  {
    if (typeof(#[1])=="int" || typeof(#[1])=="number")
    {
      addPD = int(#[1]);
    }
    if (size(#)>1)
    {
      if (typeof(#[2])=="int" || typeof(#[2])=="number")
      {
        whichengine = int(#[2]);
      }
      if (size(#)>2)
      {
        if (typeof(#[3])=="int" || typeof(#[3])=="number")
        {
          stdsum = int(#[3]);
        }
      }
    }
  }
  list b = bfctengine(f,1,whichengine,addPD,stdsum,0,0,0,0,0);
  return(b);
}
example
{
  "EXAMPLE:"; echo = 2;
  ring r = 0,(x,y),dp;
  poly f = x^2+y^3+x*y^2;
  bfctAnn(f);
  def R = reiffen(4,5); setring R;
  RC; // the Reiffen curve in 4,5
  bfctAnn(RC,0,1);
}

/*
static proc hardexamples ()
{
  //  some hard examples
  ring r1 = 0,(x,y,z,w),dp;
  // ab34
  poly ab34 = (z3+w4)*(3z2x+4w3y);
  bfct(ab34);
  // ha3
  poly ha3 = xyzw*(x+y)*(x+z)*(x+w)*(y+z+w);
  bfct(ha3);
  // ha4
  poly ha4 = xyzw*(x+y)*(x+z)*(x+w)*(y+z)*(y+w);
  bfct(ha4);
  // chal4: reiffen(4,5)*reiffen(5,4)
  ring r2 = 0,(x,y),dp;
  poly chal4 = (x4+xy4+y5)*(x5+x4y+y4);
  bfct(chal4);
  // (xy+z)*reiffen(4,5)
  ring r3 = 0,(x,y,z),dp;
  poly xyzreiffen45 = (xy+z)*(y4+yz4+z5);
  bfct(xyzreiffen45);

// sparse ideal as suggested by Alex; gives 1 as std
ideal I1 = 28191*y^2+14628*x*Dy, 24865*x^2+24072*x*Dx+17756*Dy^2;
std(I1);
}
*/