source: git/Singular/LIB/polymake.lib @ 61fbaf

//////////////////////////////////////////////////////////////////////////////
version="version polymake.lib 4.3.1.1 Sep_2022 ";
category="Tropical Geometry";
info="
LIBRARY:   polymake.lib    Computations with polytopes and fans,
                           interface to TOPCOM
AUTHOR:    Thomas Markwig,  email: keilen@mathematik.uni-kl.de
           Yue Ren,         email: ren@mathematik.uni-kl.de

WARNING:
   Most procedures will not work unless polymake or topcom is installed and
   if so, they will only work with the operating system LINUX!
   For more detailed information see IMPORTANT NOTE respectively consult the
   help string of the procedures.

   The conventions used in this library for polytopes and fans, e.g. the
   length and labeling of their vertices resp. rays, differs from the conventions
   used in polymake and thus from the conventions used in the polymake
   extension polymake.so of Singular. We recommend to use the newer polymake.so
   whenever possible.

IMPORTANT NOTE:
@texinfo
   Even though this is a Singular library for computing polytopes and fans
   such as the Newton polytope or the Groebner fan of a polynomial, most of
   the hard computations are NOT done by Singular but by the program
@* - TOPCOM by Joerg Rambau, Universitaet Bayreuth (see
     @uref{http://www.uni-bayreuth.de/de/team/rambau_joerg/TOPCOM/});
@* this library should rather be seen as an interface which allows to use a
   (very limited) number of options which topcom offers
   to compute with polytopes and fans and to make the results available in
   Singular for further computations;
   moreover, the user familiar with Singular does not have to learn the syntax
   of topcom, if the options offered here are sufficient for his
   purposes.
@* Note, though, that the procedures concerned with planar polygons are
   independent of topcom.
@end texinfo

PROCEDURES USING TOPCOM:
  triangulations()    computes all triangulations of a marked polytope
  secondaryPolytope() computes the secondary polytope of a marked polytope

PROCEDURES CONCERNED WITH PLANAR POLYGONS:
  cycleLength()    computes the cycleLength of cycle
  splitPolygon()   splits a marked polygon into vertices, facets, interior points
  eta()            computes the eta-vector of a triangulation
  findOrientedBoundary()  computes the boundary of a convex hull
  cyclePoints()    computes lattice points connected to some lattice point
  latticeArea()    computes the lattice area of a polygon
  picksFormula()   computes the ingrediants of Pick's formula for a polygon
  ellipticNF()     computes the normal form of an elliptic polygon

KEYWORDS:    polytope; fan; secondary fan; secondary polytope;
             Newton polytope; Groebner fan
";

////////////////////////////////////////////////////////////////////////////////
/// Auxiliary Static Procedures in this Library
////////////////////////////////////////////////////////////////////////////////
/// - scalarproduct
/// - intmatcoldelete
/// - intmatconcat
/// - sortlist
/// - minInList
/// - stringdelete
/// - abs
/// - commondenominator
/// - maxPosInIntvec
/// - maxPosInIntmat
/// - sortintvec
/// - matrixtointmat
////////////////////////////////////////////////////////////////////////////////

////////////////////////////////////////////////////////////////////////////////
LIB "polylib.lib";
LIB "linalg.lib";
LIB "random.lib";
////////////////////////////////////////////////////////////////////////////////

static proc mod_init()
{
  intvec save=option(get);
  option(noredefine);
  LIB "customstd.so";
  LIB "gfanlib.so";
  load("polymake.so","try");
  option(set,save);
}

///////////////////////////////////////////////////////////////////////////////
/// PROCEDURES USING TOPCOM
///////////////////////////////////////////////////////////////////////////////

proc triangulations (list polygon,list #)
"USAGE:  triangulations(polygon[,#]); list polygon, list #
ASSUME:  polygon is a list of integer vectors of the same size representing
         the affine coordinates of the lattice points
PURPOSE: the procedure considers the marked polytope given as the convex hull of
         the lattice points and with these lattice points as markings; it then
         computes all possible triangulations of this marked polytope
RETURN:  list, each entry corresponds to one triangulation and the ith entry is
               itself a list of integer vectors of size three, where each integer
               vector defines one triangle in the triangulation by telling which
               points of the input are the vertices of the triangle
NOTE:- the procedure calls for its computations the program points2triangs
       from the program topcom by Joerg Rambau, Universitaet Bayreuth; it
       therefore is necessary that this program is installed in order to use
       this  procedure; see http://www.rambau.wm.uni-bayreuth.de/TOPCOM/);
@*   - if you only want to have the regular triangulations the procedure should
       be called with the string 'regular' as optional argument
@*   - the procedure creates the files /tmp/triangulationsinput and
       /tmp/triangulationsoutput;
       the former is used as input for points2triangs and the latter is its
       output containing the triangulations of corresponding to points in the
       format of points2triangs; if you wish to use this for further
       computations with topcom, you have to call the procedure with the
       string 'keepfiles' as optional argument
@*   - note that an integer i in an integer vector representing a triangle
       refers to the ith lattice point, i.e. polygon[i]; this convention is
       different from TOPCOM's convention, where i would refer to the i-1st
       lattice point
EXAMPLE: example triangulations;   shows an example"
{
  int i,j;
  // check for optional arguments
  int regular,keepfiles;
  if (size(#)>0)
  {
    for (i=1;i<=size(#);i++)
    {
      if (typeof(#[i])=="string")
      {
        if (#[i]=="keepfiles")
        {
          keepfiles=1;
        }
        if (#[i]=="regular")
        {
          regular=1;
        }
      }
    }
  }
  // prepare the input for points2triangs by writing the input polygon in the
  // necessary format
  string spi="[";
  for (i=1;i<=size(polygon);i++)
  {
    polygon[i][size(polygon[i])+1]=1;
    spi=spi+"["+string(polygon[i])+"]";
    if (i<size(polygon))
    {
      spi=spi+",";
    }
  }
  spi=spi+"]";
  write(":w /tmp/triangulationsinput",spi);
  // call points2triangs
  if (regular==1) // compute only regular triangulations
  {
    system("sh","cd /tmp; points2triangs --regular < triangulationsinput > triangulationsoutput");
  }
  else // compute all triangulations
  {
    system("sh","cd /tmp; points2triangs < triangulationsinput > triangulationsoutput");
  }
  string p2t=read("/tmp/triangulationsoutput"); // takes result of points2triangs
  // delete the tmp-files, if no second argument is given
  if (keepfiles==0)
  {
    system("sh","cd /tmp; rm -f triangulationsinput; rm -f triangulationsoutput");
  }
  // preprocessing of p2t if points2triangs is version >= 0.15
  // brings p2t to the format of version 0.14
  string np2t; // takes the triangulations in Singular format
  for (i=1;i<=size(p2t)-2;i++)
  {
    if ((p2t[i]==":") and (p2t[i+1]=="=") and (p2t[i+2]=="["))
    {
      np2t=np2t+p2t[i]+p2t[i+1];
      i=i+3;
      while (p2t[i]!=":")
      {
        i=i+1;
      }
    }
    else
    {
      if ((p2t[i]=="]") and (p2t[i+1]==";"))
      {
        np2t=np2t+p2t[i+1];
        i=i+1;
      }
      else
      {
        np2t=np2t+p2t[i];
      }
    }
  }
  if (p2t[size(p2t)-1]=="]")
  {
    np2t=np2t+p2t[size(p2t)];
  }
  else
  {
    if (np2t[size(np2t)]!=";")
    {
      np2t=np2t+p2t[size(p2t)-1]+p2t[size(p2t)];
    }
  }
  p2t=np2t;
  np2t="";
  // transform the points2triangs output of version 0.14 into Singular format
  for (i=1;i<=size(p2t);i++)
  {
    if (p2t[i]=="=")
    {
      np2t=np2t+p2t[i]+"list(";
      i++;
    }
    else
    {
      if (p2t[i]!=":")
      {
        if ((p2t[i]=="}") and (p2t[i+1]=="}"))
        {
          np2t=np2t+"))";
          i++;
        }
        else
        {
          if (p2t[i]=="{")
          {
            np2t=np2t+"intvec(";
          }
          else
          {
            if (p2t[i]=="}")
            {
              np2t=np2t+")";
            }
            else
            {
              if (p2t[i]=="[")
              {
                // in Topcom version 17.4 (and maybe also in earlier versions) the list
                // of triangulations is indexed starting with index 0, in Singular
                // we have to start with index 1
                np2t=np2t+p2t[i]+"1+";
              }
              else
              {
                np2t=np2t+p2t[i];
              }
            }
          }
        }
      }
    }
  }
  list T;
  execute(np2t);
  // depending on the version of Topcom, the list T has or has not an entry T[1]
  // if it has none, the entry should be removed
  while (typeof(T[1])=="none")
  {
    T=delete(T,1);
  }
  // raise each index by one
  for (i=1;i<=size(T);i++)
  {
    for (j=1;j<=size(T[i]);j++)
    {
      T[i][j]=T[i][j]+1;
    }
  }
  return(T);
}
example
{
   "EXAMPLE:";
   echo=2;
   // the lattice points of the unit square in the plane
   list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
   // the triangulations of this lattice point configuration are computed
   list triang=triangulations(polygon);
   triang;
}

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

proc secondaryPolytope (list polygon,list #)
"USAGE:  secondaryPolytope(polygon[,#]); list polygon, list #
ASSUME:  - polygon is a list of integer vectors of the same size representing
           the affine coordinates of lattice points
@*       - if the triangulations of the corresponding polygon have already been
           computed with the procedure triangulations then these can be given as
           a second (optional) argument in order to avoid doing this computation
           again
PURPOSE: the procedure considers the marked polytope given as the convex hull of
         the lattice points and with these lattice points as markings; it then
         computes the lattice points of the secondary polytope given by this
         marked polytope which correspond to the triangulations computed by
         the procedure triangulations
RETURN:  list, say L, such that:
@*             L[1] = intmat, each row gives the affine coordinates of a lattice
                      point in the secondary polytope given by the marked
                      polytope corresponding to polygon
@*             L[2] = the list of corresponding triangulations
NOTE: if the triangluations are not handed over as optional argument the
      procedure calls for its computation of these triangulations the program
      points2triangs from the program topcom by Joerg Rambau, Universitaet
      Bayreuth; it therefore is necessary that this program is installed in
      order to use this procedure; see
@*    http://www.rambau.wm.uni-bayreuth.de/TOPCOM/);
EXAMPLE: example secondaryPolytope;   shows an example"
{
  // compute the triangulations of the point configuration with points2triangs
  if (size(#)==0)
  {
    list triangs=triangulations(polygon);
  }
  else
  {
    list triangs=#;
  }
  int i,j,k,l;
  intmat N[2][2]; // is used to compute areas of triangles
  intvec vertex;  // stores a point in the secondary polytope as
                  // intermediate result
  int eintrag;
  int halt;
  intmat secpoly[size(triangs)][size(polygon)];   // stores all lattice points
                                                  // of the secondary polytope
  // consider each triangulation and compute the corresponding point
  // in the secondary polytope
  for (i=1;i<=size(triangs);i++)
  {
    // for each triangulation we have to compute the coordinates
    // corresponding to each marked point
    for (j=1;j<=size(polygon);j++)
    {
      eintrag=0;
      // for each marked point we have to consider all triangles in the
      // triangulation which involve this particular point
      for (k=1;k<=size(triangs[i]);k++)
      {
        halt=0;
        for (l=1;(l<=3) and (halt==0);l++)
        {
          if (triangs[i][k][l]==j)
          {
            halt=1;
            N[1,1]=polygon[triangs[i][k][3]][1]-polygon[triangs[i][k][1]][1];
            N[1,2]=polygon[triangs[i][k][2]][1]-polygon[triangs[i][k][1]][1];
            N[2,1]=polygon[triangs[i][k][3]][2]-polygon[triangs[i][k][1]][2];
            N[2,2]=polygon[triangs[i][k][2]][2]-polygon[triangs[i][k][1]][2];
            eintrag=eintrag+abs(det(N));
          }
        }
      }
      vertex[j]=eintrag;
    }
    secpoly[i,1..size(polygon)]=vertex;
  }
  return(list(secpoly,triangs));
}
example
{
   "EXAMPLE:";
   echo=2;
   // the lattice points of the unit square in the plane
   list polygon=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
   // the secondary polytope of this lattice point configuration is computed
   list secpoly=secondaryPolytope(polygon);
   // the points in the secondary polytope
   print(secpoly[1]);
   // the corresponding triangulations
   secpoly[2];
}


////////////////////////////////////////////////////////////////////////////////
/// PROCEDURES CONCERNED WITH PLANAR POLYGONS
////////////////////////////////////////////////////////////////////////////////

proc cycleLength (list boundary,intvec interior)
"USAGE:  cycleLength(boundary,interior); list boundary, intvec interior
ASSUME:  boundary is a list of integer vectors describing a cycle in some
         convex lattice polygon around the lattice point interior ordered
         clock wise
RETURN:  string, the cycle length of the corresponding cycle in the dual
                 tropical curve
EXAMPLE: example cycleLength;   shows an example"
{
  int j;
  // create a ring whose variables are indexed by the points in
  // boundary resp. by interior
 string rst="(u"+string(interior[1])+string(interior[2]);
 for (j=1;j<=size(boundary);j++)
 {
   rst=rst+",u"+string(boundary[j][1])+string(boundary[j][2]);
 }
 rst=rst+")";
 ring cyclering = create_ring(0, rst, "lp");
  // add the first and second point at the end of boundary
  boundary[size(boundary)+1]=boundary[1];
  boundary[size(boundary)+1]=boundary[2];
  poly cl,summand; // takes the cycle length
  matrix N1[2][2]; // used to compute the area of a triangle
  matrix N2[2][2]; // used to compute the area of a triangle
  matrix N3[2][2]; // used to compute the area of a triangle
  // for each original point in boundary compute its contribution to the cycle
  for (j=2;j<=size(boundary)-1;j++)
  {
    N1=boundary[j-1]-interior,boundary[j]-interior;
    N2=boundary[j]-interior,boundary[j+1]-interior;
    N3=boundary[j+1]-interior,boundary[j-1]-interior;
    execute("summand=-u"+string(boundary[j][1])+string(boundary[j][2])+"+u"+string(interior[1])+string(interior[2])+";");
    summand=summand*(det(N1)+det(N2)+det(N3))/(det(N1)*det(N2));
    cl=cl+summand;
  }
  return(string(cl));
}
example
{
   "EXAMPLE:";
   echo=2;
   // the integer vectors in boundary are lattice points on the boundary
   // of a convex lattice polygon in the plane
   list boundary=intvec(0,0),intvec(0,1),intvec(0,2),intvec(2,2),
                 intvec(2,1),intvec(2,0);
   // interior is a lattice point in the interior of this lattice polygon
   intvec interior=1,1;
   // compute the general cycle length of a cycle of the corresponding cycle
   // in the dual tropical curve, note that (0,1) and (2,1) do not contribute
   cycleLength(boundary,interior);
}

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

proc splitPolygon (list markings)
"USAGE:  splitPolygon (markings);  markings list
ASSUME:  markings is a list of integer vectors representing lattice points in
         the plane which we consider as the marked points of the convex lattice
         polytope spanned by them
PURPOSE: split the marked points in the vertices, the points on the facets
         which are not vertices, and the interior points
RETURN:  list, L consisting of three lists
@*             L[1]    : represents the vertices the polygon ordered clockwise
@*                       L[1][i][1] = intvec, the coordinates of the ith vertex
@*                       L[1][i][2] = int, the position of L[1][i][1] in markings
@*             L[2][i] : represents the lattice points on the facet of the
                         polygon with endpoints L[1][i] and L[1][i+1]
                         (i considered modulo size(L[1]))
@*                       L[2][i][j][1] = intvec, the coordinates of the jth
                                                 lattice point on that facet
@*                       L[2][i][j][2] = int, the position of L[2][i][j][1]
                                              in markings
@*             L[3]    : represents the interior lattice points of the polygon
@*                       L[3][i][1] = intvec, coordinates of ith interior point
@*                       L[3][i][2] = int, the position of L[3][i][1] in markings
EXAMPLE: example splitPolygon;   shows an example"
{
  list vert; // stores the result
  // compute the boundary of the polygon in an oriented way
  list pb=findOrientedBoundary(markings);
  // the vertices are just the second entry of pb
  vert[1]=pb[2];
  int i,j,k;      // indices
  list boundary;  // stores the points on the facets of the
                  // polygon which are not vertices
  // append to the boundary points as well as to the vertices
  // the first vertex a second time
  pb[1]=pb[1]+list(pb[1][1]);
  pb[2]=pb[2]+list(pb[2][1]);
  // for each vertex find all points on the facet of the polygon with this vertex
  // and the next vertex as endpoints
  int z=2;
  for (i=1;i<=size(vert[1]);i++)
  {
    j=1;
    list facet; // stores the points on this facet which are not vertices
    // while the next vertex is not reached, store the boundary lattice point
    while (pb[1][z]!=pb[2][i+1])
    {
      facet[j]=pb[1][z];
      j++;
      z++;
    }
    // store the points on the ith facet as boundary[i]
    boundary[i]=facet;
    kill facet;
    z++;
  }
  // store the information on the boundary in vert[2]
  vert[2]=boundary;
  // find the remaining points in the input which are not on
  // the boundary by checking
  // for each point in markings if it is contained in pb[1]
  list interior=markings;
  for (i=size(interior);i>=1;i--)
  {
    for (j=1;j<=size(pb[1])-1;j++)
    {
      if (interior[i]==pb[1][j])
      {
        interior=delete(interior,i);
        j=size(pb[1]);
      }
    }
  }
  // store the interior points in vert[3]
  vert[3]=interior;
  // add to each point in vert the index which it gets from
  // its position in the input markings;
  // do so for ver[1]
  for (i=1;i<=size(vert[1]);i++)
  {
    j=1;
    while (markings[j]!=vert[1][i])
    {
      j++;
    }
    vert[1][i]=list(vert[1][i],j);
  }
  // do so for ver[2]
  for (i=1;i<=size(vert[2]);i++)
  {
    for (k=1;k<=size(vert[2][i]);k++)
    {
      j=1;
      while (markings[j]!=vert[2][i][k])
      {
        j++;
      }
      vert[2][i][k]=list(vert[2][i][k],j);
    }
  }
  // do so for ver[3]
  for (i=1;i<=size(vert[3]);i++)
  {
    j=1;
    while (markings[j]!=vert[3][i])
    {
      j++;
    }
    vert[3][i]=list(vert[3][i],j);
  }
  return(vert);
}
example
{
   "EXAMPLE:";
   echo=2;
   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
   // with all integer points as markings
   list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),
                intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),
                intvec(0,2),intvec(0,3);
   // split the polygon in its vertices, its facets and its interior points
   list sp=splitPolygon(polygon);
   // the vertices
   sp[1];
   // the points on facets which are not vertices
   sp[2];
   // the interior points
   sp[3];
}


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

proc eta (list triang,list polygon)
"USAGE:  eta(triang,polygon);  triang, polygon list
ASSUME:  polygon has the format of the output of splitPolygon, i.e. it is a
         list with three entries describing a convex lattice polygon in the
         following way:
@*       polygon[1] : is a list of lists; for each i the entry polygon[1][i][1]
                      is a lattice point which is a vertex of the lattice
                      polygon, and polygon[1][i][2] is an integer assigned to
                      this lattice point as identifying index
@*       polygon[2] : is a list of lists; for each vertex of the polygon,
                      i.e. for each entry in polygon[1], it contains a list
                      polygon[2][i], which contains the lattice points on the
                      facet with endpoints polygon[1][i] and polygon[1][i+1]
                      - i considered mod size(polygon[1]);
                      each such lattice point contributes an entry
                      polygon[2][i][j][1] which is an integer
                      vector giving the coordinate of the lattice point and an
                      entry polygon[2][i][j][2] which is the identifying index
@*       polygon[3] : is a list of lists, where each entry corresponds to a
                      lattice point in the interior of the polygon, with
                      polygon[3][j][1] being the coordinates of the point
                      and polygon[3][j][2] being the identifying index;
@*       triang is a list of integer vectors all of size three describing a
         triangulation of the polygon described by polygon; if an entry of
         triang is the vector (i,j,k) then the triangle is built by the vertices
         with indices i, j and k
RETURN:  intvec, the integer vector eta describing that vertex of the Newton
                 polytope discriminant of the polygon whose dual cone in the
                 Groebner fan contains the cone of the secondary fan of the
                 polygon corresponding to the given triangulation
NOTE:  for a better description of eta see Gelfand, Kapranov,
       Zelevinski: Discriminants, Resultants and multidimensional Determinants.
       Chapter 10.
EXAMPLE: example eta;   shows an example"
{
  int i,j,k,l,m,n; // index variables
  list ordpolygon;   // stores the lattice points in the order
                     // used in the triangulation
  list triangarea; // stores the areas of the triangulations
  intmat N[2][2];  // used to compute triangle areas
  // 1) store the lattice points in the order used in the triangulation
  // go first through all vertices of the polytope
  for (j=1;j<=size(polygon[1]);j++)
  {
    ordpolygon[polygon[1][j][2]]=polygon[1][j][1];
  }
  // then consider all inner points
  for (j=1;j<=size(polygon[3]);j++)
  {
    ordpolygon[polygon[3][j][2]]=polygon[3][j][1];
  }
  // finally consider all lattice points on the boundary which are not vertices
  for (j=1;j<=size(polygon[2]);j++)
  {
    for (i=1;i<=size(polygon[2][j]);i++)
    {
      ordpolygon[polygon[2][j][i][2]]=polygon[2][j][i][1];
    }
  }
  // 2) compute for each triangle in the triangulation the area of the triangle
  for (i=1;i<=size(triang);i++)
  {
    // Note that the ith lattice point in orderedpolygon has the
    // number i-1 in the triangulation!
    N=ordpolygon[triang[i][1]]-ordpolygon[triang[i][2]],ordpolygon[triang[i][1]]-ordpolygon[triang[i][3]];
    triangarea[i]=abs(det(N));
  }
  intvec ETA;        // stores the eta_ij
  int etaij;         // stores the part of eta_ij during computations
                     // which comes from triangle areas
  int seitenlaenge;  // stores the part of eta_ij during computations
                     // which comes from boundary facets
  list seiten;       // stores the lattice points on facets of the polygon
  intvec v;          // used to compute a facet length
  // 3) store first in seiten[i] all lattice points on the facet
  //    connecting the ith vertex,
  //    i.e. polygon[1][i], with the i+1st vertex, i.e. polygon[1][i+1],
  //    where we replace i+1
  //    1 if i=size(polygon[1]);
  //    then append the last entry of seiten once more at the very
  //    beginning of seiten, so
  //    that the index is shifted by one
  for (i=1;i<=size(polygon[1]);i++)
  {
    if (i<size(polygon[1]))
    {
      seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][i+1]);
    }
    else
    {
      seiten[i]=list(polygon[1][i])+polygon[2][i]+list(polygon[1][1]);
    }
  }
  seiten=insert(seiten,seiten[size(seiten)],0);
  // 4) compute the eta_ij for all vertices of the polygon
  for (j=1;j<=size(polygon[1]);j++)
  {
    // the vertex itself contributes a 1
    etaij=1;
    // check for each triangle in the triangulation ...
    for (k=1;k<=size(triang);k++)
    {
      // ... if the vertex is actually a vertex of the triangle ...
      if ((polygon[1][j][2]==triang[k][1]) or (polygon[1][j][2]==triang[k][2]) or (polygon[1][j][2]==triang[k][3]))
      {
        // ... if so, add the area of the triangle to etaij
        etaij=etaij+triangarea[k];
        // then check if that triangle has a facet which is contained
        // in one of the
        // two facets of the polygon which are adjecent to the given vertex ...
        // these two facets are seiten[j] and seiten[j+1]
        for (n=j;n<=j+1;n++)
        {
          // check for each lattice point in the facet of the polygon ...
          for (l=1;l<=size(seiten[n]);l++)
          {
            // ... and for each lattice point in the triangle ...
            for (m=1;m<=size(triang[k]);m++)
            {
              // ... if they coincide and are not the vertex itself ...
              if ((seiten[n][l][2]==triang[k][m]) and (seiten[n][l][2]!=polygon[1][j][2]))
              {
                // if so, then compute the vector pointing from this
                // lattice point to the vertex
                v=polygon[1][j][1]-seiten[n][l][1];
                // and the lattice length of this vector has to be
                // subtracted from etaij
                etaij=etaij-abs(gcd(v[1],v[2]));
              }
            }
          }
        }
      }
    }
    // store etaij in the list
    ETA[polygon[1][j][2]]=etaij;
  }
  // 5) compute the eta_ij for all lattice points on the facets
  //    of the polygon which are not vertices, these are the
  //    lattice points in polygon[2][1] to polygon[2][size(polygon[1])]
  for (i=1;i<=size(polygon[2]);i++)
  {
    for (j=1;j<=size(polygon[2][i]);j++)
    {
      // initialise etaij
      etaij=0;
      // initialise seitenlaenge
      seitenlaenge=0;
      // check for each triangle in the triangulation ...
      for (k=1;k<=size(triang);k++)
      {
        // ... if the vertex is actually a vertex of the triangle ...
        if ((polygon[2][i][j][2]==triang[k][1]) or (polygon[2][i][j][2]==triang[k][2]) or (polygon[2][i][j][2]==triang[k][3]))
        {
          // ... if so, add the area of the triangle to etaij
          etaij=etaij+triangarea[k];
          // then check if that triangle has a facet which is contained in the
          // facet of the polygon which contains the lattice point in question,
          // this is the facet seiten[i+1];
          // check for each lattice point in the facet of the polygon ...
          for (l=1;l<=size(seiten[i+1]);l++)
          {
            // ... and for each lattice point in the triangle ...
            for (m=1;m<=size(triang[k]);m++)
            {
              // ... if they coincide and are not the vertex itself ...
              if ((seiten[i+1][l][2]==triang[k][m]) and (seiten[i+1][l][2]!=polygon[2][i][j][2]))
              {
                // if so, then compute the vector pointing from
                // this lattice point to the vertex
                v=polygon[2][i][j][1]-seiten[i+1][l][1];
                // and the lattice length of this vector contributes
                // to seitenlaenge
                seitenlaenge=seitenlaenge+abs(gcd(v[1],v[2]));
              }
            }
          }
        }
      }
      // if the lattice point was a vertex of any triangle
      // in the triangulation ...
      if (etaij!=0)
      {
        // then eta_ij is the sum of the triangle areas minus seitenlaenge
        ETA[polygon[2][i][j][2]]=etaij-seitenlaenge;
      }
      else
      {
        // otherwise it is just zero
        ETA[polygon[2][i][j][2]]=0;
      }
    }
  }
  // 4) compute the eta_ij for all inner lattice points of the polygon
  for (j=1;j<=size(polygon[3]);j++)
  {
    // initialise etaij
    etaij=0;
    // check for each triangle in the triangulation ...
    for (k=1;k<=size(triang);k++)
    {
      // ... if the vertex is actually a vertex of the triangle ...
      if ((polygon[3][j][2]==triang[k][1]) or (polygon[3][j][2]==triang[k][2]) or (polygon[3][j][2]==triang[k][3]))
      {
        // ... if so, add the area of the triangle to etaij
        etaij=etaij+triangarea[k];
      }
    }
    // store etaij in ETA
    ETA[polygon[3][j][2]]=etaij;
  }
  return(ETA);
}
example
{
   "EXAMPLE:";
   echo=2;
   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
   // with all integer points as markings
   list polygon=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),
                intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),
                intvec(0,2),intvec(0,3);
   // split the polygon in its vertices, its facets and its interior points
   list sp=splitPolygon(polygon);
   // define a triangulation by connecting the only interior point
   //        with the vertices
   list triang=intvec(1,2,5),intvec(1,5,10),intvec(1,5,10);
   // compute the eta-vector of this triangulation
   eta(triang,sp);
}

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

proc findOrientedBoundary (list polygon)
"USAGE: findOrientedBoundary(polygon); polygon list
ASSUME: polygon is a list of integer vectors defining integer lattice points
        in the plane
RETURN: list l with the following interpretation
@*            l[1] = list of integer vectors such that the polygonal path
                     defined by these is the boundary of the convex hull of
                     the lattice points in polygon
@*            l[2] = list, the redundant points in l[1] have been removed
EXAMPLE:     example findOrientedBoundary;   shows an example"
{
  // Order the vertices such that passing from one to the next we travel along
  // the boundary of the convex hull of the vertices clock wise
  int d,k,i,j;
  intmat D[2][2];
  /////////////////////////////////////
  // Treat first the pathological cases that the polygon is not two-dimensional:
  /////////////////////////////////////
  // if the polygon is empty or only one point or a line segment of two points
  if (size(polygon)<=2)
  {
    return(list(polygon,polygon));
  }
  // check is the polygon is only a line segment given by more than two points;
  // for this first compute sum of the absolute values of the determinants
  // of the matrices whose
  // rows are the vectors pointing from the first to the second point
  // and from the
  // the first point to the ith point for i=3,...,size(polygon);
  // if this sum is zero
  // then the polygon is a line segment and we have to find its end points
  d=0;
  for (i=3;i<=size(polygon);i++)
  {
    D=polygon[2]-polygon[1],polygon[i]-polygon[1];
    d=d+abs(det(D));
  }
  if (d==0) // then polygon is a line segment
  {
    intmat laenge[size(polygon)][size(polygon)];
    intvec mp;
    //   for this collect first all vectors pointing from one lattice
    //   point to the next,
    //   compute their pairwise angles and their lengths
    for (i=1;i<=size(polygon)-1;i++)
    {
      for (j=i+1;j<=size(polygon);j++)
      {
        mp=polygon[i]-polygon[j];
        laenge[i,j]=abs(gcd(mp[1],mp[2]));
      }
    }
    mp=maxPosInIntmat(laenge);
    list endpoints=polygon[mp[1]],polygon[mp[2]];
    intvec abstand;
    for (i=1;i<=size(polygon);i++)
    {
      abstand[i]=0;
      if (i<mp[1])
      {
        abstand[i]=laenge[i,mp[1]];
      }
      if (i>mp[1])
      {
        abstand[i]=laenge[mp[1],i];
      }
    }
    polygon=sortlistbyintvec(polygon,abstand);
    return(list(polygon,endpoints));
  }
  ///////////////////////////////////////////////////////////////
  list orderedvertices;  // stores the vertices in an ordered way
  list minimisedorderedvertices;  // stores the vertices in an ordered way;
                                  // redundant ones removed
  list comparevertices; // stores vertices which should be compared to
                        // the testvertex
  orderedvertices[1]=polygon[1]; // set the starting vertex
  minimisedorderedvertices[1]=polygon[1]; // set the starting vertex
  intvec testvertex=polygon[1];  //vertex to which the others have to be compared
  intvec startvertex=polygon[1]; // keep the starting vertex to test,
                                 // when the end is reached
  int endtest;                   // is set to one, when the end is reached
  int startvertexfound;// is 1, once for some testvertex a candidate
                       // for the next vertex has been found
  polygon=delete(polygon,1);    // delete the testvertex
  intvec v,w;
  int l=1;  // counts the vertices
  // the basic idea is that a vertex can be
  // the next one on the boundary if all other vertices
  // lie to the right of the vector v pointing
  // from the testvertex to this one; this can be tested
  // by checking if the determinant of the 2x2-matrix
  // with first column v and second column the vector w,
  // pointing from the testvertex to the new vertex,
  // is non-positive; if this is the case for all
  // new vertices, then the one in consideration is
  // a possible choice for the next vertex on the boundary
  // and it is stored in naechste; we can then order
  // the candidates according to their distance from
  // the testvertex; then they occur on the boundary in that order!
  while (endtest==0)
  {
    list naechste;  // stores the possible choices for the next vertex
    k=1;
    for (i=1;i<=size(polygon);i++)
    {
      d=0;  // stores the value of the determinant of (v,w)
      v=polygon[i]-testvertex; // points from the testvertex to the ith vertex
      comparevertices=delete(polygon,i); // we needn't compare v to itself
      // we should compare v to the startvertex-testvertex;
      // in the first calling of the loop
      // this is irrelevant since the difference will be zero;
      // however, later on it will
      // be vital, since we delete the vertices
      // which we have already tested from the list
      // of all vertices, and when all vertices
      // on the boundary have been found we would
      // therefore find a vertex in the interior
      // as candidate; but always testing against
      // the starting vertex, this cannot happen
      comparevertices[size(comparevertices)+1]=startvertex;
      for (j=1;(j<=size(comparevertices)) and (d<=0);j++)
      {
        w=comparevertices[j]-testvertex; // points form the testvertex
                                         // to the jth vertex
        D=v,w;
        d=det(D);
      }
      if (d<=0) // if all determinants are non-positive,
      { // then the ith vertex is a candidate
        naechste[k]=list(polygon[i],i,scalarproduct(v,v));// we store the vertex,
                                                          //its position, and its
        k++; // distance from the testvertex
      }
    }
    if (size(naechste)>0) // then a candidate for the next vertex has been found
    {
      startvertexfound=1; // at least once a candidate has been found
      naechste=sortlist(naechste,3);  // we order the candidates according
                                      // to their distance from testvertex;
      for (j=1;j<=size(naechste);j++) // then we store them in this
      { // order in orderedvertices
        l++;
        orderedvertices[l]=naechste[j][1];
      }
      testvertex=naechste[size(naechste)][1];  // we store the last one as
                                               // next testvertex;
      // store the next corner of NSD
      minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex;
      naechste=sortlist(naechste,2); // then we reorder the vertices
                                     // according to their position
      for (j=size(naechste);j>=1;j--) // and we delete them from the vertices
      {
        polygon=delete(polygon,naechste[j][2]);
      }
    }
    else // that means either that the vertex was inside the polygon,
    {    // or that we have reached the last vertex on the boundary
         // of the polytope
      if (startvertexfound==0) // the vertex was in the interior;
      { // we delete it and start all over again
        orderedvertices[1]=polygon[1];
        minimisedorderedvertices[1]=polygon[1];
        testvertex=polygon[1];
        startvertex=polygon[1];
        polygon=delete(polygon,1);
      }
      else // we have reached the last vertex on the boundary of
      { // the polytope and can stop
        endtest=1;
      }
    }
    kill naechste;
  }
  // test if the first vertex in minimisedorderedvertices
  // is on the same line with the second and
  // the last, i.e. if we started our search in the
  // middle of a face; if so, delete it
  v=minimisedorderedvertices[2]-minimisedorderedvertices[1];
  w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1];
  D=v,w;
  if (det(D)==0)
  {
    minimisedorderedvertices=delete(minimisedorderedvertices,1);
  }
  // test if the first vertex in minimisedorderedvertices
  // is on the same line with the two
  // last ones, i.e. if we started our search at the end of a face;
  // if so, delete it
  v=minimisedorderedvertices[size(minimisedorderedvertices)-1]-minimisedorderedvertices[1];
  w=minimisedorderedvertices[size(minimisedorderedvertices)]-minimisedorderedvertices[1];
  D=v,w;
  if (det(D)==0)
  {
    minimisedorderedvertices=delete(minimisedorderedvertices,size(minimisedorderedvertices));
  }
  return(list(orderedvertices,minimisedorderedvertices));
}
example
{
   "EXAMPLE:";
   echo=2;
// the following lattice points in the plane define a polygon
   list polygon=intvec(0,0),intvec(3,1),intvec(1,0),intvec(2,0),
                intvec(1,1),intvec(3,2),intvec(1,2),intvec(2,3),
                intvec(2,4);
// we compute its boundary
   list boundarypolygon=findOrientedBoundary(polygon);
// the points on the boundary ordered clockwise are boundarypolygon[1]
   boundarypolygon[1];
// the vertices of the boundary are boundarypolygon[2]
   boundarypolygon[2];
}


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

proc cyclePoints (list triang,list points,int pt)
"USAGE:      cyclePoints(triang,points,pt)  triang,points list, pt int
ASSUME:      - points is a list of integer vectors describing the lattice
               points of a marked polygon;
@*           - triang is a list of integer vectors describing a triangulation
               of the marked polygon in the sense that an integer vector of
               the form (i,j,k) describes the triangle formed by polygon[i],
               polygon[j] and polygon[k];
@*           - pt is an integer between 1 and size(points), singling out a
               lattice point among the marked points
PURPOSE:     consider the convex lattice polygon, say P, spanned by all lattice
             points in points which in the triangulation triang are connected
             to the point points[pt]; the procedure computes all marked points
             in points which lie on the boundary of that polygon, ordered
             clockwise
RETURN:      list, of integer vectors which are the coordinates of the lattice
                   points on the boundary of the above mentioned polygon P, if
                   this polygon is not the empty set (that would be the case if
                   points[pt] is not a vertex of any triangle in the
                   triangulation); otherwise return the empty list
EXAMPLE:     example cyclePoints;   shows an example"
{
  int i,j; // indices
  list v;  // saves the indices of lattice points connected to the
           // interior point in the triangulation
  // save all points in triangulations containing pt in v
  for (i=1;i<=size(triang);i++)
  {
    if ((triang[i][1]==pt) or (triang[i][2]==pt) or (triang[i][3]==pt))
    {
      j++;
      v[3*j-2]=triang[i][1];
      v[3*j-1]=triang[i][2];
      v[3*j]=triang[i][3];
    }
  }
  if (size(v)==0)
  {
    return(list());
  }
  // remove pt itself and redundancies in v
  for (i=size(v);i>=1;i--)
  {
    j=1;
    while ((j<i) and (v[i]!=v[j]))
    {
      j++;
    }
    if ((j<i) or (v[i]==pt))
    {
      v=delete(v,i);
    }
  }
  // save in pts the coordinates of the points with indices in v
  list pts;
  for (i=1;i<=size(v);i++)
  {
    pts[i]=points[v[i]];
  }
  // consider the convex polytope spanned by the points in pts,
  // find the points on the
  // boundary and order them clockwise
  return(findOrientedBoundary(pts)[1]);
}
example
{
   "EXAMPLE:";
   echo=2;
   // the lattice polygon spanned by the points (0,0), (3,0) and (0,3)
   // with all integer points as markings
   list points=intvec(1,1),intvec(3,0),intvec(2,0),intvec(1,0),
               intvec(0,0),intvec(2,1),intvec(0,1),intvec(1,2),
               intvec(0,2),intvec(0,3);
   // define a triangulation
   list triang=intvec(1,2,5),intvec(1,5,7),intvec(1,7,9),intvec(8,9,10),
               intvec(1,8,9),intvec(1,2,8);
   // compute the points connected to (1,1) in triang
   cyclePoints(triang,points,1);
}

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

proc latticeArea (list polygon)
"USAGE:  latticeArea(polygon);   polygon list
ASSUME:  polygon is a list of integer vectors in the plane
RETURN:  int, the lattice area of the convex hull of the lattice points in
              polygon, i.e. twice the Euclidean area
EXAMPLE: example polygonlatticeArea;   shows an example"
{
  list pg=findOrientedBoundary(polygon)[2];
  int area;
  intmat M[2][2];
  for (int i=2;i<=size(pg)-1;i++)
  {
    M[1,1..2]=pg[i]-pg[1];
    M[2,1..2]=pg[i+1]-pg[1];
    area=area+abs(det(M));
  }
  return(area);
}
example
{
   "EXAMPLE:";
   echo=2;
   // define a polygon with lattice area 5
   list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1),
                intvec(2,1),intvec(0,0);
   latticeArea(polygon);
}

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

proc picksFormula (list polygon)
"USAGE:  picksFormula(polygon);   polygon list
ASSUME:  polygon is a list of integer vectors in the plane and consider their
         convex hull C
RETURN:  list, L of three integersthe
@*             L[1] : the lattice area of C, i.e. twice the Euclidean area
@*             L[2] : the number of lattice points on the boundary of C
@*             L[3] : the number of interior lattice points of C
NOTE: the integers in L are related by Pick's formula, namely: L[1]=L[2]+2*L[3]-2
EXAMPLE: example picksFormula;   shows an example"
{
  list pg=findOrientedBoundary(polygon)[2];
  int area,bdpts,i;
  intmat M[2][2];
  // compute the lattice area of the polygon, i.e. twice the Euclidean area
  for (i=2;i<=size(pg)-1;i++)
  {
    M[1,1..2]=pg[i]-pg[1];
    M[2,1..2]=pg[i+1]-pg[1];
    area=area+abs(det(M));
  }
  // compute the number of lattice points on the boundary
  intvec edge;
  pg[size(pg)+1]=pg[1];
  for (i=1;i<=size(pg)-1;i++)
  {
    edge=pg[i]-pg[i+1];
    bdpts=bdpts+abs(gcd(edge[1],edge[2]));
  }
  // Pick's formula says that the lattice area A, the number g of interior
  // points and
  // the number b of boundary points are connected by the formula: A=b+2g-2
  return(list(area,bdpts,(area-bdpts+2) div 2));
}
example
{
   "EXAMPLE:";
   echo=2;
   // define a polygon with lattice area 5
   list polygon=intvec(1,2),intvec(1,0),intvec(2,0),intvec(1,1),
                intvec(2,1),intvec(0,0);
   list pick=picksFormula(polygon);
   // the lattice area of the polygon is:
   pick[1];
   // the number of lattice points on the boundary is:
   pick[2];
   // the number of interior lattice points is:
   pick[3];
   // the number's are related by Pick's formula:
   pick[1]-pick[2]-2*pick[3]+2;
}

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

proc ellipticNF (list polygon)
"USAGE:  ellipticNF(polygon);   polygon list
ASSUME:  polygon is a list of integer vectors in the plane such that their
         convex hull C has precisely one interior lattice point; i.e. C is the
         Newton polygon of an elliptic curve
PURPOSE: compute the normal form of the polygon with respect to the unimodular
         affine transformations T=A*x+v; there are sixteen different normal forms
         (see e.g. Bjorn Poonen, Fernando Rodriguez-Villegas: Lattice Polygons
                   and the number 12.  Amer. Math. Monthly  107  (2000),  no. 3,
                   238--250.)
RETURN:  list, L such that
@*             L[1] : list whose entries are the vertices of the normal form of
                      the polygon
@*             L[2] : the matrix A of the unimodular transformation
@*             L[3] : the translation vector v of the unimodular transformation
@*             L[4] : list such that the ith entry is the image of polygon[i]
                      under the unimodular transformation T
EXAMPLE: example ellipticNF;   shows an example"
{
  int i;            // index
  intvec edge;      // stores the vector of an edge
  intvec boundary;  // stores lattice lengths of the edges of the Newton cycle
  // find the vertices of the Newton cycle and order it clockwise
  list pg=findOrientedBoundary(polygon)[2];
  // check if there is precisely one interior point in the Newton polygon
  if (picksFormula(pg)[3]!=1)
  {
    ERROR("The polygon has not precisely one interior point!");
  }
  // insert the first vertex at the end once again
  pg[size(pg)+1]=pg[1];
  // compute the number of lattice points on each edge
  for (i=1;i<=size(pg)-1;i++)
  {
    edge=pg[i]-pg[i+1];
    boundary[i]=1+abs(gcd(edge[1],edge[2]));
  }
  // store the values of boundary once more adding the first two at the end
  intvec tboundary=boundary,boundary[1],boundary[2];
  // sort boundary in an asecending way
  intvec sbd=sortintvec(boundary);
  // find the first edge having the maximal number of lattice points
  int max=maxPosInIntvec(boundary);
  // some computations have to be done over the rationals
  ring transformationring=0,x,lp;
  intvec trans;    // stores the vector by which we have to translate the polygon
  intmat A[2][2];  // stores the matrix by which we have to transform the polygon
  matrix M[3][3];  // stores the projective coordinates of the points
                   // which are to be transformed
  matrix N[3][3];  // stores the projective coordinates of the points to
                   // which M is to be transformed
  intmat T[3][3];  // stores the unimodular affine transformation in
                   // projective form
  // add the second point of pg once again at the end
  pg=insert(pg,pg[2],size(pg));
  // if there is only one edge which has the maximal number of lattice points,
  // then M should be:
  M=pg[max],1,pg[max+1],1,pg[max+2],1;
  // consider the 16 different cases which can occur:
  // Case 1:
  if (sbd==intvec(2,2,2))
  {
    N=0,1,1,1,2,1,2,0,1;
  }
  // Case 2:
  if (sbd==intvec(2,2,3))
  {
    N=2,0,1,0,0,1,1,2,1;
  }
  // Case 3:
  if (sbd==intvec(2,3,4))
  {
    // here the orientation of the Newton polygon is important !
    if (tboundary[max+1]==3)
    {
      N=3,0,1,0,0,1,0,2,1;
    }
    else
    {
      N=0,0,1,3,0,1,0,2,1;
    }
  }
  // Case 4:
  if (sbd==intvec(3,3,5))
  {
    N=4,0,1,0,0,1,0,2,1;
  }
  // Case 5:
  if (sbd==intvec(4,4,4))
  {
    N=3,0,1,0,0,1,0,3,1;
  }
  // Case 6+7:
  if (sbd==intvec(2,2,2,2))
  {
    // there are two different polygons which has four edges all of length 2,
    // but only one of them has two edges whose direction vectors form a matrix
    // of determinant 3
    A=pg[1]-pg[2],pg[3]-pg[2];
    while ((max<4) and (det(A)!=3))
    {
      max++;
      A=pg[max]-pg[max+1],pg[max+2]-pg[max+1];
    }
    // Case 6:
    if (det(A)==3)
    {
      M=pg[max],1,pg[max+1],1,pg[max+2],1;
      N=1,0,1,0,2,1,2,1,1;
    }
    // Case 7:
    else
    {
      N=2,1,1,1,0,1,0,1,1;
    }
  }
  // Case 8:
  if (sbd==intvec(2,2,2,3))
  {
    // the orientation of the polygon is important
    A=pg[max]-pg[max+1],pg[max+2]-pg[max+1];
    if (det(A)==2)
    {
      N=2,0,1,0,0,1,0,1,1;
    }
    else
    {
      N=0,0,1,2,0,1,1,2,1;
    }
  }
  // Case 9:
  if (sbd==intvec(2,2,3,3))
  {
    // if max==1, then the 5th entry in tboundary is the same as the first
    if (max==1)
    {
      max=5;
    }
    // if boundary=3,2,2,3 then set max=4
    if (tboundary[max+1]!=3)
    {
      max=4;
    }
    M=pg[max],1,pg[max+1],1,pg[max+2],1;
    // the orientation of the polygon matters
    A=pg[max-1]-pg[max],pg[max+1]-pg[max];
    if (det(A)==4)
    {
      N=2,0,1,0,0,1,0,2,1;
    }
    else
    {
      N=0,2,1,0,0,1,2,0,1;
    }
  }
  // Case 10:
  if (sbd==intvec(2,2,3,4))
  {
    // the orientation of the polygon matters
    if (tboundary[max+1]==3)
    {
      N=3,0,1,0,0,1,0,2,1;
    }
    else
    {
      N=0,0,1,3,0,1,2,1,1;
    }
  }
  // Case 11:
  if (sbd==intvec(2,3,3,4))
  {
    N=3,0,1,0,0,1,0,2,1;
  }
  // Case 12:
  if (sbd==intvec(3,3,3,3))
  {
    N=2,0,1,0,0,1,0,2,1;
  }
  // Case 13:
  if (sbd==intvec(2,2,2,2,2))
  {
    // compute the angles of the polygon vertices
    intvec dt;
    for (i=1;i<=5;i++)
    {
      A=pg[i]-pg[i+1],pg[i+2]-pg[i+1];
      dt[i]=det(A);
    }
    dt[6]=dt[1];
    // find the vertex to be mapped to (0,1)
    max=1;
    while ((dt[max]!=2) or (dt[max+1]!=2))
    {
      max++;
    }
    M=pg[max],1,pg[max+1],1,pg[max+2],1;
    N=0,1,1,1,2,1,2,1,1;
  }
  // Case 14:
  if (sbd==intvec(2,2,2,2,3))
  {
    N=2,0,1,0,0,1,0,1,1;
  }
  // Case 15:
  if (sbd==intvec(2,2,2,3,3))
  {
    // find the vertex to be mapped to (2,0)
    if (tboundary[max+1]!=3)
    {
      max=5;
      M=pg[max],1,pg[max+1],1,pg[max+2],1;
    }
    N=2,0,1,0,0,1,0,2,1;
  }
  // Case 16:
  if (sbd==intvec(2,2,2,2,2,2))
  {
    N=2,0,1,1,0,1,0,1,1;
  }
  // we have to transpose the matrices M and N
  M=transpose(M);
  N=transpose(N);
  // compute the unimodular affine transformation, which is of the form
  // A11 A12 | T1
  // A21 A22 | T2
  //  0   0  | 1
  T=matrixtointmat(N*inverse(M));
  // the upper-left 2x2-block is A
  A=T[1..2,1..2];
  // the upper-right 2x1-block is the translation vector
  trans=T[1,3],T[2,3];
  // transform now the lattice points of the polygon with respect to A and T
  list nf;
  for (i=1;i<=size(polygon);i++)
  {
    intmat V[2][1]=polygon[i];
    V=A*V;
    nf[i]=intvec(V[1,1]+trans[1],V[2,1]+trans[2]);
    kill V;
  }
  return(list(findOrientedBoundary(nf)[2],A,trans,nf));
}
example
{
   "EXAMPLE:";
   echo=2;
   ring r=0,(x,y),dp;
   // the Newton polygon of the following polynomial
   //     has precisely one interior point
   poly f=x22y11+x19y10+x17y9+x16y9+x12y7+x9y6+x7y5+x2y3;
   list polygon=list(intvec(22,11),intvec(19,10),intvec(17,9),
                     intvec(16,9), intvec(12,7),intvec(9,6),
                     intvec(7,5),intvec(2,3));
   // its lattice points are
   polygon;
   // find its normal form
   list nf=ellipticNF(polygon);
   // the vertices of the normal form are
   nf[1];
   // it has been transformed by the unimodular affine transformation A*x+v
   // with matrix A
   nf[2];
   // and translation vector v
   nf[3];
   // the 3rd lattice point ...
   polygon[3];
   // ... has been transformed to
   nf[4][3];
}


/////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////////
/// AUXILIARY PROCEDURES, WHICH ARE DECLARED STATIC
/////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////////
/// - scalarproduct
/// - intmatcoldelete
/// - intmatconcat
/// - sortlist
/// - minInList
/// - stringdelete
/// - abs
/// - commondenominator
/// - maxPosInIntvec
/// - maxPosInIntmat
/// - sortintvec
/// - matrixtointmat
/////////////////////////////////////////////////////////////////////////////////

static proc scalarproduct (intvec w,intvec v)
"USAGE:      scalarproduct(w,v); w,v intvec
ASSUME:      w and v are integer vectors of the same length
RETURN:      int, the scalarproduct of v and w
NOTE:        the procedure is called by findOrientedBoundary"
{
  int sp;
  for (int i=1;i<=size(w);i++)
  {
    sp=sp+v[i]*w[i];
  }
  return(sp);
}

static proc intmatcoldelete (def w,int i)
"USAGE:      intmatcoldelete(w,i); w intmat, i int
RETURN:      intmat, the integer matrix w with the ith comlumn deleted
NOTE:        the procedure is called by intmatsort"
{
  if (typeof(w)=="intmat")
  {
    if ((i<1) or (i>ncols(w)) or (ncols(w)==1))
    {
      return(w);
    }
    if (i==1)
    {
      intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),2..ncols(w)];
      return(M);
    }
    if (i==ncols(w))
    {
      intmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),1..ncols(w)-1];
      return(M);
    }
    else
    {
      intmat M[nrows(w)][i-1]=w[1..nrows(w),1..i-1];
      intmat N[nrows(w)][ncols(w)-i]=w[1..nrows(w),i+1..ncols(w)];
      return(intmatconcat(M,N));
    }
  }
  if (typeof(w)=="bigintmat")
  {
    if ((i<1) or (i>ncols(w)) or (ncols(w)==1))
    {
      return(w);
    }
    if (i==1)
    {
      bigintmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),2..ncols(w)];
      return(M);
    }
    if (i==ncols(w))
    {
      bigintmat M[nrows(w)][ncols(w)-1]=w[1..nrows(w),1..ncols(w)-1];
      return(M);
    }
    else
    {
      bigintmat MN[nrows(w)][ncols(w)-1];
      MN[1..nrows(w),1..i-1]=w[1..nrows(w),1..i-1];
      MN[1..nrows(w),i..ncols(w)-1]=w[1..nrows(w),i+1..ncols(w)];
      return(MN);
    }
  } else
  {
    ERROR("intmatcoldelete: input matrix has to be of type intmat or bigintmat");
    intmat M; return(M);
  }
}

static proc intmatconcat (intmat M,intmat N)
"USAGE:      intmatconcat(M,N); M,N intmat
RETURN:      intmat, M and N concatenated
NOTE:        the procedure is called by intmatcoldelete and sortintmat"
{
  if (nrows(M)>=nrows(N))
  {
    int m=nrows(M);

  }
  else
  {
    int m=nrows(N);
  }
  intmat P[m][ncols(M)+ncols(N)];
  P[1..nrows(M),1..ncols(M)]=M[1..nrows(M),1..ncols(M)];
  P[1..nrows(N),ncols(M)+1..ncols(M)+ncols(N)]=N[1..nrows(N),1..ncols(N)];
  return(P);
}

static proc sortlist (list v,int pos)
"USAGE:      sortlist(v,pos); v list, pos int
RETURN:      list, the list L ordered in an ascending way according to the pos-th entries
NOTE:        called by tropicalCurve"
{
  if(size(v)==1)
  {
    return(v);
  }
  list w=minInList(v,pos);
  v=delete(v,w[2]);
  v=sortlist(v,pos);
  v=list(w[1])+v;
  return(v);
}

static proc minInList (list v,int pos)
"USAGE:      minInList(v,pos); v list, pos int
RETURN:      list, (v[i],i) such that v[i][pos] is minimal
NOTE:        called by sortlist"
{
  int min=v[1][pos];
  int minpos=1;
  for (int i=2;i<=size(v);i++)
  {
    if (v[i][pos]<min)
    {
      min=v[i][pos];
      minpos=i;
    }
  }
  return(list(v[minpos],minpos));
}

static proc stringdelete (string w,int i)
"USAGE:      stringdelete(w,i); w string, i int
RETURN:      string, the string w with the ith component deleted
NOTE:        the procedure is called by texnumber and choosegfanvector"
{
  if ((i>size(w)) or (i<=0))
  {
    return(w);
  }
  if ((size(w)==1) and (i==1))
  {
    return("");

  }
  if (i==1)
  {
    return(w[2..size(w)]);
  }
  if (i==size(w))
  {
    return(w[1..size(w)-1]);
  }
  else
  {
    string erg=w[1..i-1],w[i+1..size(w)];
    return(erg);
  }
}

static proc abs (def n)
"USAGE:      abs(n); n poly or int
RETURN:      poly or int, the absolute value of n"
{
  if (n>=0)
  {
    return(n);
  }
  else
  {
    return(-n);
  }
}

static proc commondenominator (matrix M)
"USAGE:   commondenominator(M);  M matrix
ASSUME:   the base ring has characteristic zero
RETURN:   int, the lowest common multiple of the denominators of the leading coefficients
               of the entries in M
NOTE:        the procedure is called from polymakeToIntmat"
{
  int i,j;
  int kgV=1;
  // successively build the lowest common multiple of the denominators of the leading coefficients
  // of the entries in M
  for (i=1;i<=nrows(M);i++)
  {
    for (j=1;j<=ncols(M);j++)
    {
      kgV=lcm(kgV,int(denominator(leadcoef(M[i,j]))));
    }
  }
  return(kgV);
}

static proc maxPosInIntvec (intvec v)
"USAGE:      maxPosInIntvec(v); v intvec
RETURN:      int, the first position of a maximal entry in v
NOTE:        called by sortintmat"
{
  int max=v[1];
  int maxpos=1;
  for (int i=2;i<=size(v);i++)
  {
    if (v[i]>max)
    {
      max=v[i];
      maxpos=i;
    }
  }
  return(maxpos);
}

static proc maxPosInIntmat (intmat v)
"USAGE:      maxPosInIntmat(v); v intmat
ASSUME:      v has a unique maximal entry
RETURN:      intvec, the position (i,j) of the maximal entry in v
NOTE:        called by findOrientedBoundary"
{
  int max=v[1,1];
  intvec maxpos=1,1;
  int i,j;
  for (i=1;i<=nrows(v);i++)
  {
    for (j=1;j<=ncols(v);j++)
    {
      if (v[i,j]>max)
      {
        max=v[i,j];
        maxpos=i,j;
      }
    }
  }
  return(maxpos);
}

static proc sortintvec (intvec w)
"USAGE:      sortintvec(v); v intvec
RETURN:      intvec, the entries of v are ordered in an ascending way
NOTE:        called from ellipticNF"
{
  int j,k,stop;
  intvec v=w[1];
  for (j=2;j<=size(w);j++)
  {
    k=1;
    stop=0;
    while ((k<=size(v)) and (stop==0))
    {
      if (v[k]<w[j])
      {
        k++;
      }
      else
      {
        stop=1;
      }
    }
    if (k==size(v)+1)
    {
      v=v,w[j];
    }
    else
    {
      if (k==1)
      {
        v=w[j],v;
      }
      else
      {
        v=v[1..k-1],w[j],v[k..size(v)];
      }
    }
  }
  return(v);
}

static proc sortlistbyintvec (list L,intvec w)
"USAGE:      sortlistbyintvec(L,w); L list, w intvec
RETURN:      list, the entries of L are ordered such that the corresponding reordering of
                   w would order w in an ascending way
NOTE:        called from ellipticNF"
{
  int j,k,stop;
  intvec v=w[1];
  list LL=L[1];
  for (j=2;j<=size(w);j++)
  {
    k=1;
    stop=0;
    while ((k<=size(v)) and (stop==0))
    {
      if (v[k]<w[j])
      {
        k++;
      }
      else
      {
        stop=1;
      }
    }
    if (k==size(v)+1)
    {
      v=v,w[j];
      LL=insert(LL,L[j],size(LL));
    }
    else
    {
      if (k==1)
      {
        v=w[j],v;
        LL=insert(LL,L[j]);
      }
      else
      {
        v=v[1..k-1],w[j],v[k..size(v)];
        LL=insert(LL,L[j],k-1);
      }
    }
  }
  return(LL);
}

static proc matrixtointmat (matrix MM)
"USAGE:      matrixtointmat(v); MM matrix
ASSUME:      MM is a matrix with only integers as entries
RETURN:      intmat, the matrix MM has been transformed to type intmat
NOTE:        called from ellipticNF"
{
  intmat M[nrows(MM)][ncols(MM)]=M;
  int i,j;
  for (i=1;i<=nrows(M);i++)
  {
    for (j=1;j<=ncols(M);j++)
    {
      execute("M["+string(i)+","+string(j)+"]="+string(MM[i,j])+";");
    }
  }
  return(M);
}

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

static proc polygonToCoordinates (list points)
"USAGE:      polygonToCoordinates(points);   points list
ASSUME:      points is a list of integer vectors each of size two describing the
             marked points of a convex lattice polygon like the output of
             polygonDB
RETURN:      list, the first entry is a string representing the coordinates
                   corresponding to the latticpoints separated by commata
                   the second entry is a list where the ith entry is a string
                   representing the coordinate of corresponding to the ith
                   lattice point the third entry is the latex format of the
                   first entry
NOTE:        the procedure is called by fan"
{
  string coord;
  list coords;
  string latex;
  for (int i=1;i<=size(points);i++)
  {
    coords[i]="u"+string(points[i][1])+string(points[i][2]);
    coord=coord+coords[i]+",";
    latex=latex+"u_{"+string(points[i][1])+string(points[i][2])+"},";
  }
  coord=coord[1,size(coord)-1];
  latex=latex[1,size(latex)-1];
  return(list(coord,coords,latex));
}

static proc intmatAddFirstColumn (def M,string art)
"USAGE:  intmatAddFirstColumn(M,art);  M intmat, art string
ASSUME:  - M is an integer matrix where a first column of 0's or 1's should be added
@*       - art is one of the following strings:
@*         + 'rays'   : indicating that a first column of 0's should be added
@*         + 'points' : indicating that a first column of 1's should be added
RETURN:  intmat, a first column has been added to the matrix"
{
  if (typeof (M) == "intmat")
  {
    intmat N[nrows(M)][ncols(M)+1];
    int i,j;
    for (i=1;i<=nrows(M);i++)
    {
      if (art=="rays")
      {
        N[i,1]=0;
      }
      else
      {
        N[i,1]=1;
      }
      for (j=1;j<=ncols(M);j++)
      {
        N[i,j+1]=M[i,j];
      }
    }
    return(N);
  }
  if (typeof (M) == "bigintmat")
  {
    bigintmat N[nrows(M)][ncols(M)+1];
    int i,j;
    for (i=1;i<=nrows(M);i++)
    {
      if (art=="rays")
      {
        N[i,1]=0;
      }
      else
      {
        N[i,1]=1;
      }
      for (j=1;j<=ncols(M);j++)
      {
        N[i,j+1]=M[i,j];
      }
    }
    return(N);
  }
  else
  {
    ERROR ("intmatAddFirstColumn: input matrix has to be either intmat or bigintmat");
    intmat N;
    return (N);
  }
}