source: git/Singular/LIB/reesclos.lib @ f8c998c

////////////////////////////////////////////////////////////////////////////
version="version reesclos.lib 4.0.0.0 Jun_2013 "; // $Id$
category="Commutative Algebra";

info="
LIBRARY:     reesclos.lib   PROCEDURES TO COMPUTE THE INT. CLOSURE OF AN IDEAL
AUTHOR:      Tobias Hirsch, email: hirsch@math.tu-cottbus.de
             Janko Boehm, email: boehm@mathematik.uni-kl.de
             Magdaleen Marais, email: magdaleen@aims.ac.za

OVERVIEW:
 A library to compute the integral closure of an ideal I in a polynomial ring
 R=k[x(1),...,x(n)] using the Rees Algebra R[It] of I. It computes the integral
 closure of R[It],
 which is a graded subalgebra of R[t]. The degree-k-component is the integral
 closure of the k-th power of I.

 In contrast to the previous version, the library uses 'normal.lib' to compute the 
 integral closure of R[It]. This improves the performance considerably.

PROCEDURES:
 ReesAlgebra(I);        computes the Rees Algebra of an ideal I
 normalI(I[,p[,r]]);    computes the integral closure of an ideal I using R[It]
";

LIB "normal.lib";       // for HomJJ
LIB "standard.lib";     // for groebner

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

proc ReesAlgebra (ideal I)
"USAGE:    ReesAlgebra (I); I = ideal
RETURN:   The Rees algebra R[It] as an affine ring, where I is an ideal in R.
          The procedure returns a list containing two rings:
          [1]: a ring, say RR; in the ring an ideal ker such that R[It]=RR/ker

          [2]: a ring, say Kxt; the basering with additional variable t
               containing an ideal mapI that defines the map RR-->Kxt
EXAMPLE:  example ReesAlgebra; shows an example
"
{
  // remember the data of the basering

  def oldring = basering;
  string oldchar = charstr(basering);
  string oldvar  = varstr(basering);
  string oldord  = ordstr(basering);
  int n = ncols(I);
  ideal m = maxideal(1);


  // Create a new ring with variables for each generator of I

  execute ("ring Rees = "+oldchar+",("+oldvar+",U(1.."+string(n)+")),dp");


  // Kxt is the old ring with additional variable t
  // Here I -> t*I, so the generators of I generate the subalgebra R[It] in Kxt

  execute ("ring Kxt = "+oldchar+",("+oldvar+",t),dp");
  ideal I = fetch(oldring,I);
  ideal m = fetch(oldring,m);
  int k;
  for (k=1;k<=n;k++)
  {
    I[k]=t*I[k];
  }


  // Now we map from Rees to Kxt, identity on the original variables, and
  // U(k) -> I[k]

  ideal mapI = m,I;
  map phi = Rees,mapI;
  ideal zero = 0;
  export (mapI);

  // Now the Rees-Algebra is Rees/ker(phi)

  setring Rees;
  ideal ker = preimage(Kxt,phi,zero);
  export (ker);

  list result = Rees,Kxt;

  return(result);

}
example
{
  "EXAMPLE:"; echo=2;
  ring R = 0,(x,y),dp;
  ideal I = x2,xy4,y5;
  list L = ReesAlgebra(I);
  def Rees = L[1];       // defines the ring Rees, containing the ideal ker
  setring Rees;          // passes to the ring Rees
  Rees;
  ker;                   // R[It] is isomorphic to Rees/ker
}


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

static
proc ClosureRees (list L)
"USAGE:    ClosureRees (L); L a list
ASSUME:   L is a list containing
          - a ring L[1], inside L[1] an ideal ker such that L[1]/ker is
            isomorphic to the Rees Algebra R[It] of an ideal I in k[x]
          - a ring L[2]=k[x,t], inside L[1] an ideal mapI defining the
            map L[1] --> L[2] with image R[It]
RETURN:   quotients of elements of k[x,t] representing generators of the
          integral closure of R[It]. The result of ClosureRees is a list
          images, the first size(images)-1 entries are the numerators of the
          generators, the last one is the universal denominator
"
{
  int dblvl=printlevel-voice+2;   // toggles how much data is printed
                                  // during the procedure

  def Kxt = basering;
  def R(1) = L[1];
  setring R(1);                   // declaration of variables used later
  ideal ker(1)=ker;               // in STEP 2
  list nor = normal(ker);
  list preimages=nor[2];
  setring Kxt;
  map psi=R(1),mapI;              // from ReesAlgebra: the map Rees->Kxt
  ideal images=(psi(preimages))[1];
  ideal psii = images[size(images)]*ideal(psi);
  list imagesl = images[1..size(images)];
  list psil =psii[1..size(psii)];
  imagesl=psil+imagesl;
  return(imagesl);
}


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

static
proc ClosurePower(list images, list #)
"USAGE:    ClosurePower (L [,#]); L a list, # an optional list containing an
          integer
ASSUME:   - L is a list containing generators of the closure of R[It] in k[x,t]
            (the first size(L)-1 elements are the numerators, the last one
            is the denominator)
          - if # is given: #[1] is an integer, compute generators for the
                           closure of I, I^2, ..., I^#[1]
RETURN:   the integral closure of I, ... I^#[1]. If # is not given, compute
          the closure of all powers up to the maximum degree in t occurring
          in the closure of R[It] (so this is the last power whose closure is
          not just the sum/product of the smaller powers). The returned
          result is a list of elements of k[x,t] containing generators of the
          closure of the desired powers of I. "
{
  int dblvl=printlevel-voice+2;   // toggles how much data is printed
                                  // during the procedure

  int j,k,d,computepow;                    // some counters
  int pow=0;
  int length = size(images)-1;             // the number of generators
  poly image;
  poly @denominator = images[length+1];     // the universal denominator

  if (size(#)>0)
  {
    pow=#[1];
  }
  computepow=pow;

  if (dblvl>0)
  {
    "";
    "// The generators of the closure of R[It]:";
  }

  intmat m[nvars(basering)-1][1];  // an intvec used for jet and maxdeg1
  intvec tw=m,1;                   // such that t has weight 1 and all
                                   // other variables have weight 0

  // Construct the generators of the closure of R[It] as elements of k[x,t]
  // If # is not given, determine the highest degree pow in t that occurs.

  for (j=1;j<=length;j++)
  {
    images[j] = (images[j]/@denominator); // construct the fraction
    image = images[j];
    if (dblvl>0)
    {
      "generator",j,":",image;
    }

    if (computepow==0)              // #[1] not given or ==0 => compute pow
    {
      if (maxdeg1(image,tw)>pow)    // from poly.lib
      {
        pow=maxdeg1(image,tw);
      }
    }
  }

  if (dblvl>0)
  {
    "";
    if (computepow==0)
    {
      "// Compute the closure up to the given powers of I";
    }
    else
    {
     "// Compute the closure up to the maximal power of t that occured:",pow;
    }
  }

  // Construct a list consisting of #[1] resp. pow times the zero ideal

  ideal CurrentPower=0;
  list result;
  for (k=1;k<=pow;k++)
  {
    result=insert(result,CurrentPower);
  }

  // For each generator and each k, add its degree-k-coefficient to the #
  // closure of I^k

  for (j=1;j<=length;j++)
  {
    for (k=1;k<=pow;k++)
    {
      image=images[j]-jet(images[j],k-1,tw);
      if (image<>0)
      {
        image=subst(image/t^k,t,0);
        if (image<>0)
        {
          result[k]=result[k]+image;
        }
      }
    }
  }

  if (dblvl>0)
  {
    "";
    "// The 'pure' parts of degrees 1..pow:";
    result;
    "";
  }

  // finally, add the suitable products of generators in lower degrees

  for (k=2;k<=pow;k++)
  {
    for (j=1;j<=(k div 2);j++)
    {
      result[k]=result[k]+result[j]*result[k-j];
    }
  }

  return(result);
}

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

proc normalI(ideal I, list #)
"USAGE:    normalI (I [,p[,r]]); I an ideal, p and r optional integers
RETURN:   the integral closure of I, ..., I^p, where I is an ideal in the
          polynomial ring R=k[x(1),...x(n)]. If p is not given, or p==0,
          compute the closure of all powers up to the maximum degree in t
          occurring in the closure of R[It] (so this is the last power whose
          closure is not just the sum/product of the smaller). If r
          is given and r==1, normalI starts with a check whether I is already a
          radical ideal.
          The result is a list containing the closure of the desired powers of
          I as ideals of the basering.
DISPLAY:  The procedure displays more comments for higher printlevel.
EXAMPLE:  example normalI; shows an example
"
{
  int dblvl=printlevel-voice+2;   // toggles how much data is printed
                                  // during the procedure

  def BAS=basering;               // remember the basering

  // two simple cases: principal ideals and radical ideals are always
  // integrally closed

  if (size(I)<=1)        // includes the case I=(0)
  {
    if (dblvl>0)
    {
      "// Trivial case: I is a principal ideal";
    }
    list result=I;
    if (size(#)>0)
    {
      for (int k=1;k<=#[1]-1;k++)
      {
        result=insert(result,I*result[k],k);
      }
    }
    return(result);
  }

  int testrad=0;      // do the radical check?
  if (size(#)>1)
  {
    testrad=#[2];
  }

  if (testrad==1)
  {
    if (dblvl>0)
    {
      "//Check whether I is radical";
    }

    if (size(reduce(radical(I),std(I)))==0)
    {
      if (dblvl>0)
      {
        "//Trivial case: I is a radical ideal";
      }
      list result=I;
      if (size(#)>0)
      {
        for (int k=1;k<=#[1]-1;k++)
        {
          result=insert(result,I*result[k],k);
        }
      }
      return(result);
    }
  }

  // start with the computation of the Rees Algebra R[It] of I

  if (dblvl>0)
  {
    "// We start with the Rees Algebra of I:";
  }

  list Rees = ReesAlgebra(I);
  def R(1)=Rees[1];
  def Kxt=Rees[2];
  setring R(1);

  if (dblvl>0)
  {
    R(1);
    ker;
    "";
    "// Now ClosureRees computes generators for the integral closure";
    "// of R[It] step by step";
  }

  // ClosureRees computes fractions in R[x,t] representing the generators
  // of the closure of R[It] in k[x,t], which is the same as the closure
  // in Q(R[It]).
  // the first size(images)-1 entries are the numerators of the gene-
  // rators, the last entry is the 'universal' denominator

  setring Kxt;
  list images = ClosureRees(Rees);

  // ClosureRees was done after the first HomJJ-call
  // ==> I is integrally closed, and images consists of the only entry "closed"

  if ((size(images)==1) && (typeof(images[1])=="string"))
  {
    if (dblvl>0)
    {
      "//I is integrally closed!";
    }

    setring BAS;
    list result=I;
    if (size(#)>0)
    {
      for (int k=1;k<=#[1]-1;k++)
      {
        result=insert(result,I*result[k],k);
      }
    }
    return(result);
  }

  // construct the fractions corresponding to the generators of the
  // closure of I and its powers, depending on # (in fact, they will
  // not be real fractions, of course). This is done in ClosurePower.
  list result = ClosurePower(images,#);

  // finally fetch the result to the old basering

  setring BAS;
  list result=fetch(Kxt,result);
  return(result);
}
example
{
  "EXAMPLE:"; echo=2;
  ring R=0,(x,y),dp;
  ideal I = x2,xy4,y5;
  list J = normalI(I);
  I;
  J;                             // J[1] is the integral closure of I
}

/*
LIB"reesclos.lib";


// 1.  x^i,y^i in k[x,y]
//     geht bis i = 19 (800sec), bis i=10 wenige Sekunden,
//     bei i = 20 ueber 1GB Hauptspeicher, in der 9. Iteration no memory
//     (braucht 20 Iterationen)

  ring r = 0,(x,y),dp;
  int i = 6;
  ideal I = x^i,y^i;
  list J = normalI(I);
  I;
  J;


//================================================================


// 2. x^i,y^i,z^i in k[x,y,z]
//    aehnlich wie 1., funktioniert aber nur bis i=5 und dauert dort
//    >1 h


//================================================================


// 3. scheitert in der ersten Iteration beim Radikal
//    Standardbasis des singulaeren Ortes: 7h (in char0),
//    in char(p) viel schneller, obwohl kleine Koeffizienten
//    schon bei Radikal -Test braucht er zu lang (>1h)

  ring r = 0,(x,y,z),dp;
  //ring r = 32003,(x,y,z),dp;

  ideal I = x2+xy3-5z,z3+y2-xzy,x2y3z5+y3-y5;
  list l= ReesAlgebra(I);
  list J = normalI(I);
  I;
  J;

*/