source: git/Singular/LIB/reesclos.lib @ b9b906

///////////////////////////////////////////////////////////////////////////////
version="$id reesclos.lib,v 1.30 2000/12/5 hirsch Exp $";
category="Commutative Algebra";
info="
LIBRARY:     reesclos.lib Rees Algebra and Integral Closure of Ideals
AUTHOR:      Tobias Hirsch, email: hirsch@math.tu-cottbus.de

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] (in the same manner as done in the library 'normal.lib');
 which is a graded subalgebra of R[t]. The degree-k-component is the integral
 closure of the k-th power of I.

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
COMPUTE:  The Rees algebra R[I*t] as an affine ring, where I is an ideal in R
RETURN:   a list containing two rings:
          [1]: a ring, say RR; in the ring an ideal ker such that R[I*t]=RR/ker
          [2]: a ring, say Kxt; the basering with additional variable t con-
               taining an ideal mapI defining 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[I*t] is isomorphic to Rees/ker
}


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

static proc ClosureRees (list L)
"USAGE:    ClosureRees (L); L 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]
COMPUTE:  quotients of elements of k[x,t] representing generators of the
          integral closure of R[It]
RETURN:   a list images, the first size(images)-1 entries are the nu-
          merators of the generators, the last one is the universal deno-
          minator
"
{
  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
  poly p;

  // some auxiliary variables

  int i=1;         // counts the number of steps to reach the closure of R(1)
  int check=0;     // a 'boolean' variable for several checks
  int isIsoSing=0; // '1' in case of an isolated singularity

  /////// STEP 1: ///////////////////////////////////////////////////////////
  // construct R(i) step by step as done in normal.lib; 2 differences:     //
  //  - since the input is a prime ideal, no splitting will occur          //
  //  - the intermediate rings and nonzerodivisors for J are remembered    //
  ///////////////////////////////////////////////////////////////////////////

  if (dblvl>0)
  {
    "// STEP 1: Compute the integral closure of R[It]";
  }

  list RS;

  while (check==0)  // repeat until the closure is reached
  {
    // construction of HomJJ, J an ideal containing the non-normal locus
    // of R(i)/ker(i), as done in normalizationPrimes in normal.lib for
    // the special case that we are working with a prime ideal

    if (dblvl>0)
    {
      "";
      "// We are in step",i;
      pause();
    }

    if (homog(ker(i))==1)
    {
      list SM=mstd(ker(i));
    }
    else
    {
      list SM=groebner(ker(i)),ker(i);
    }

    if (dblvl>0)
    {
      "// Standard basis of the current ideal:";
      SM[1];
    }

    // In the first iteration, we have to compute the singular locus and
    // to check whether it is an isolated singularity (this makes things
    // much easier: for further iterations, we just take the max. ideal).
    // If not, we still fetch the singular locus but have to compute
    // its radical.

    if (i==1)
    {
      ideal J=minor(jacob(SM[2]),nvars(basering)-dim(SM[1]));
      ideal sin=J+SM[2];
      if (homog(SM[2])==1)
      {
        list JM=mstd(sin);
      }
      else
      {
        list JM=groebner(sin),sin;
      }

      if (dim(JM[1])==0 && homog(SM[2])==1)
      {
        isIsoSing=1;
      }
      J=equiRadical(JM[2]);
    }
    else
    {
      if (isIsoSing==0)
      {
        ideal J=equiRadical(JM[2]);
      }
      else
      {
        ideal J=maxideal(1);
      }
    }

    if (dblvl>0)
    {
      "// Radical of the singular locus:";
      J;
    }

    ideal SL=simplify(reduce(JM[2],SM[1]),2);
    JM =J,J;
    poly nzd=SL[1];           // universal denominator for HomJJ

    if (dblvl>0)
    {
      "// The non-zerodivisor";
      nzd;
      pause();
    }

    list RR=SM[1],SM[2],JM[2],SL[1];
    RS=HomJJ(RR);

    if (RS[2]==1)             // we've reached the integral closure
    {
      if (dblvl>0)
      {
        "";
        "// We've reached the integral closure after",i,"iterations";
        pause();
      }
      check=1;
    }
    else                      // prepare the next iteration with new
    {                         // ring R(i+1)
      ideal MJ=JM[2];

      def R(i+1)=RS[1];       // the data of and some variable decla-
      setring R(i+1);         // rations in R(i+1) needed in STEP 2
      ideal ker(i+1)=endid;
      map phi=R(i),endphi;
      poly p;

      if (isIsoSing==0)       // fetch the singular locus if it is not
      {                       // an isolated singularity
        list JM=mstd(simplify(phi(MJ)+ker(i+1),4));
      }
      i++;
    }
  }

  if (i==1)                     // R[It] (and thus I) was integrally
  {                             // closed ==> we're already done
        list result="closed";
        return(result);
  }


  /////// STEP 2: ////////////////////////////////////////////////////////
  // compute representations of the ring variables of R(i) as fractions //
  // of elements of R(1);                                               //
  ////////////////////////////////////////////////////////////////////////

  int length=nvars(R(i));  // the number of variables of the last ring
  int j,k,n;               // some counters
  string mapstr;           // will be used while constructing preimages

  list preimages;          // here the fractions are stored in the
                           // form var(j)=preimages[j]/preimages[length+1]
                           // ('=' means identification via the inclusion)
                           // the last entry corresponds to nzd in R(i)

  // For each variable (counter j) and for each intermediate ring (k):
  // find preimages of var(j)*endphi(nzd_k-1) in R(k-1).
  // Finally, do the same for nzd.

  if (dblvl>0)
  {
    "// STEP 2: Compute fractions representing the ring variables of";
    "           the last ring";
  }

  for (j=1;j<=length+1;j++)
  {
    setring R(i);

    if (j<=length)          // do it with for ring variables...
    {
      p=var(j);
    }
    else                   // ...and finally for nzd in R(i)
    {
      p=1;
    }

    // get back from R(i) to R(1) step by step

    for (k=i;k>1;k--)
    {
      // clear the fraction in the representation in R(i)

      p=p*phi(nzd);

      // compute the preimage of [p mod ker(k)] under phi in R(k-1):
      // as p is an element of im(phi), there is a poly h such that
      // h(vars(R(k-1)) is mapped to [p mod ker (k)], and h can be com-
      // puted as the normal form of a w.r.t <ker(k),Z(n)-phi(k)(n)> in R(k)[Z]

         // compute this normal form h...

      if (j==1)    // in the first iteration: construct S(k)=R(k)[Z], fetch
                   // endphi (the ideal defining phi) and ker(k) and construct
                   // the ideal from above
      {
        execute ("ring S(k) = "+charstr(R(k))+",("+varstr(R(k))+",Z(1.."+string(ncols(endphi))+")),(dp("+string(nvars(R(k)))+"),dp("+string(ncols(endphi))+"));");
        ideal endphi = imap(R(k),endphi);
        ideal J = imap(R(k),ker(k));
        for (n=1;n<=ncols(endphi);n++)
        {
          J=J+(Z(n)-endphi[n]);
        }
        J=groebner(J);
        poly h=NF(imap(R(k),p),J);
      }
      else
      {
        setring S(k);
        h=NF(imap(R(k),p),J);
      }

         // and compute h(vars(R(k-1)))

      setring R(k-1);
      if (j==1)  // in the first iteration: compute backmap:S(k)-->R(k-1)
      {
        mapstr="map backmap = S(k),";
        for (n=1;n<=nvars(R(k));n++)
        {
          mapstr=mapstr+"0,";
        }
        execute (mapstr+"maxideal(1);");
      }

      p=NF(backmap(h),std(ker(k-1)));
    }

    // when down to R(1), store the result in the list preimages

    preimages=insert(preimages,p,j-1);
    if (dblvl>0)
    {
      if (j<=length)
      {
        "numerator of variable ",j,":",p;
      }
      else
      {
        "";
        "and finally the 'universal' denominator:",p;
      }
    }
  }

  // at the end: go back to the original basering and construct gene-
  // rators of the closure of I

  setring Kxt;
  map psi=R(1),mapI;              // from ReesAlgebra: the map Rees->Kxt

  list images=psi(preimages);

  if (dblvl>-1)
  {
    pause();
    "";
    "// Get back to the original basering and construct the";
    "// generators of the closure of I";
    "";
    "// Back in k[x,t], the fractions, stored in the list images:";
    for (int j=1;j<=size(images);j++)
    {
      if (j<size(images))
      {
        "numerator",j,":",images[j];
      }
      else
      {
        "";
        "denominator:  ",images[j];
      }
    }
    pause();
  }
  return (images);
}


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

static proc ClosurePower(list images, list #)
"USAGE:    ClosurePower (L [,#]);
          - L 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]
COMPUTE:  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 one that is not just
          the sum/product of the above ones).
RETURN:   a list containing generators of the closure of the desired powers
          of I as elements of k[x,t].
"
{
  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
COMPUTE:  the integral closure of I, ..., I^p. 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 one that
          is not just the sum/product of the above ones).
          If r is given and <>1, the check whether the input ideal is
          already a radical ideal is omitted.
RETURN:   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=1;      // do the radical check?
  if (size(#)>1)
  {
    if (#[2]<>1)
    {
      testrad=0;
    }
  }
  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 is the integral closure of I
}
///////////////////////////////////////////////////////////////////////////////
/*
  BEISPIELE
*/