source: git/Singular/LIB/polymake.lib @ e84f07d

version="$Id: polymake.lib,v 1.11 2009-03-16 11:34:10 keilen Exp $";
category="Tropical Geometry";
info="
LIBRARY:        polymake.lib    Computations with polytopes and fans, interface to polymake and TOPCOM
AUTHOR:         Thomas Markwig,  email: keilen@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.

IMPORTANT NOTE:
     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
@*   -  polymake by Ewgenij Gawrilow, TU Berlin and Michael Joswig, TU Darmstadt
@*      (see http://www.math.tu-berlin.de/polymake/),
@*   respectively (only in the procedure triangularions) by the program
@*   -  topcom by Joerg Rambau, Universitaet Bayreuth
@*      (see http://www.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM);
@*   this library should rather be seen as an interface which allows to use a (very limited)
     number of options which polymake respectively 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 polymake
     or topcom, if the options offered here are sufficient for his purposes.
@*   Note, though, that the procedures concerned with planar polygons are independent of
     both, polymake and topcom.

PROCEDURES USING POLYMAKE:
     polymakePolytope(list)            computes the vertices of a polytope using polymake
     newtonPolytope(poly)              computes the Newton polytope of the polynomial
     newtonPolytopeLP(poly)            computes the lattice points of the Newton polytope of the polynomial
     normalFan(intmat,intmat,list)     computes the normal fan of a polytope
     groebnerFan(poly)                 computes the Groebner fan of a polynomial
     intmatToPolymake(intmat,string)   transforms an integer matrix into polymake format
     polymakeToIntmat(string,string)   transforms polymake output into an integer matrix

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

PROCEDURES USING POLYMAKE AND TOPCOM:
     secondaryFan(list)            computes the secondary fan of a marked polytope

PROCEDURES CONERNED WITH PLANAR POLYGONS:
     cycleLength(list,intvec)      computes the cycleLength of cycle dual to list with interior point intvec
     splitPolygon(list)            splits a marked polygon into vertices, facets and interior points
     eta(list,list)                computes the eta-vector of a triangulation
     findOrientedBoundary(list)    computes the boundary of the convex hull of a list of lattice points
     cyclePoints(list,list,int)    computes lattice points connected to a lattice point in a triangulation
     latticeArea(list)             computes the lattice area of a polygon
     picksFormula(list)            computes the ingrediants of Pick's formula for a polygon
     ellipticNF(list)              computes the normal form of an elliptic polygon
     ellipticNFDB(int)             displays the 16 normal forms of elliptic polygons

AUXILARY PROCEDURES:
     polymakeKeepTmpFiles(int)     determines whether the files created in /tmp should be kept

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

";

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

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


////////////////////////////////////////////////////////////////////////////////////////
/// PROCEDURES USING POLYMAKE
////////////////////////////////////////////////////////////////////////////////////////

proc polymakePolytope (intmat polytope,list #)
"USAGE:  polymakePolytope(polytope[,#]);   polytope list, # string
ASSUME:  each row of polytope gives the coordinates of a lattice point of a polytope
         with their affine coordinates as given by the output of secondaryPolytope
PURPOSE: the procedure calls polymake to compute the vertices of the polytope as well
         as its dimension and information on its facets
RETURN:  list, L with four entries
@*            L[1] : an integer matrix whose rows are the coordinates of vertices
                     of the polytope
@*            L[2] : the dimension of the polytope
@*            L[3] : a list whose ith entry explains to which vertices the ith vertex
                     of the Newton polytope is connected -- i.e. L[3][i] is an integer
                     vector and an entry k in there means that the vertex L[1][i] is
                     connected to the vertex L[1][k]
@*            L[4] : an integer matrix whose rows mulitplied by (1,var(1),...,var(nvar)) give
                     a linear system of equations describing the affine hull of the polytope,
                     i.e. the smallest affine space containing the polytope
NOTE: -  for its computations the procedure calls the program polymake by Ewgenij Gawrilow,
         TU Berlin and Michael Joswig, TU Darmstadt; it therefore is necessary that
         this program is installed in order to use this procedure;
         see http://www.math.tu-berlin.de/polymake/
@*    -  note that in the vertex edge graph we have changed the polymake convention which
         starts indexing its vertices by zero while we start with one !
@*    -  the procedure creates the file  /tmp/polytope.polymake which contains the polytope
         in polymake format; if you wish to use this for further computations with polymake,
         you have to use the procedure polymakeKeepTmpFiles in before
@*    -  moreover, the procedure creates the file /tmp/polytope.output which it deletes
         again before ending
@*    -  it is possible to give as an optional second argument as string which then will be
         used instead of 'polytope' in the name of the polymake output file
EXAMPLE: example polymakePolytope;   shows an example"
{
  // the header for the file secendarypolytope.polymake
  string sp="_application polytope
_version 2.2
_type RationalPolytope

POINTS
";
  int i,j;
  // set the name for the polymake output file
  if (size(#)>0)
  {
    if (typeof(#[1])=="string")
    {
      string dateiname=#[1];
    }
    else
    {
      string dateiname="polytope";
    }
  }
  else
  {
    string dateiname="polytope";
  }
  // create the lattice point list for polymake
  sp=sp+intmatToPolymake(polytope,"points");
  // initialise dateiname.polymake and compute the vertices
  write(":w /tmp/"+dateiname+".polymake",sp);
  system("sh","cd /tmp; polymake "+dateiname+".polymake VERTICES > "+dateiname+".output");
  string vertices=read("/tmp/"+dateiname+".output");
  system("sh","/bin/rm /tmp/"+dateiname+".output");
  intmat np=polymakeToIntmat(vertices,"affine");
  // compute the dimension
  system("sh","cd /tmp; polymake "+dateiname+".polymake DIM > "+dateiname+".output");
  string pdim=read("/tmp/"+dateiname+".output");
  system("sh","/bin/rm /tmp/"+dateiname+".output");
  pdim=pdim[5,size(pdim)-6];
  execute("int nd="+pdim+";");
  // compute the vertex-edge graph
  system("sh","cd /tmp; polymake "+dateiname+".polymake GRAPH > "+dateiname+".output");
  string vertexedgegraph=read("/tmp/"+dateiname+".output");
  system("sh","/bin/rm /tmp/"+dateiname+".output");
  vertexedgegraph=vertexedgegraph[7,size(vertexedgegraph)-8];
  string newveg;
  for (i=1;i<=size(vertexedgegraph);i++)
  {
    if (vertexedgegraph[i]=="{")
    {
      newveg=newveg+"intvec(";
    }
    else
    {
      if (vertexedgegraph[i]=="}")
      {
        newveg=newveg+"),";
      }
      else
      {
        if (vertexedgegraph[i]==" ")
        {
          newveg=newveg+",";
        }
        else
        {
          newveg=newveg+vertexedgegraph[i];
        }
      }
    }
  }
  newveg=newveg[1,size(newveg)-1];
  execute("list nveg="+newveg+";");
  // raise each entry in nveg by one
  for (i=1;i<=size(nveg);i++)
  {
    for (j=1;j<=size(nveg[i]);j++)
    {
      nveg[i][j]=nveg[i][j]+1;
    }
  }
  // compute the affine hull
  system("sh","cd /tmp; polymake "+dateiname+".polymake AFFINE_HULL > "+dateiname+".output");
  string equations=read("/tmp/"+dateiname+".output");
  system("sh","/bin/rm /tmp/"+dateiname+".output");
  if (size(equations)>14)
  {
    intmat neq=polymakeToIntmat(equations,"cleardenom");
  }
  else
  {
    intmat neq[1][ncols(polytope)+1];
  }
  // delete the tmp-files, if polymakekeeptmpfiles is not set
  if (defined(polymakekeeptmpfiles)==0)
  {
    system("sh","/bin/rm /tmp/"+dateiname+".polymake");
  }
  // return the files
  return(list(np,nd,nveg,neq));
}
example
{
   "EXAMPLE:";
   echo=2;
   // the lattice points of the unit square in the plane
   list points=intvec(0,0),intvec(0,1),intvec(1,0),intvec(1,1);
   // the secondary polytope of this lattice point configuration is computed
   intmat secpoly=secondaryPolytope(points)[1];
   list np=polymakePolytope(secpoly);
   // the vertices of the secondary polytope are:
   np[1];
   // its dimension is
   np[2];
   // np[3] contains information how the vertices are connected to each other,
   // e.g. the first vertex (number 0) is connected to the second one
   np[3][1];
   // the affine hull has the equation
   ring r=0,x(1..4),dp;
   matrix M[5][1]=1,x(1),x(2),x(3),x(4);
   np[4]*M;
}

proc newtonPolytope (poly f,list #)
"USAGE: newtonPolytope(f[,#]);  f poly, # string
RETURN: list, L with four entries
@*            L[1] : an integer matrix whose rows are the coordinates of vertices
                     of the Newton polytope of f
@*            L[2] : the dimension of the Newton polytope of f
@*            L[3] : a list whose ith entry explains to which vertices the ith vertex
                     of the Newton polytope is connected -- i.e. L[3][i] is an integer
                     vector and an entry k in there means that the vertex L[1][i] is
                     connected to the vertex L[1][k]
@*            L[4] : an integer matrix whose rows mulitplied by (1,var(1),...,var(nvar)) give
                     a linear system of equations describing the affine hull of the Newton
                     polytope, i.e. the smallest affine space containing the Newton polytope
NOTE: -  if we replace the first column of L[4] by zeros, i.e. if we move the affine hull to
         the origin, then we get the equations for the orthogonal comploment of the linearity
         space of the normal fan dual to the Newton polytope, i.e. we get the EQUATIONS that
         we need as input for polymake when computing the normal fan
@*    -  the procedure calls for its computation polymake by Ewgenij Gawrilow,
         TU Berlin and Michael Joswig, so it only works if polymake is installed;
         see http://www.math.tu-berlin.de/polymake/
@*    -  the procedure creates the file  /tmp/newtonPolytope.polymake which contains the polytope
         in polymake format and which can be used for further computations with polymake
@*    -  moreover, the procedure creates the file /tmp/newtonPolytope.output which it deletes
         again before ending
@*    -  it is possible to give as an optional second argument as string which then will be
         used instead of 'newtonPolytope' in the name of the polymake output file
EXAMPLE: example newtonPolytope;   shows an example"
{
  int i,j;
  // compute the list of exponent vectors of the polynomial, which are the lattice points
  // whose convex hull is the Newton polytope of f
  intmat exponents[size(f)][nvars(basering)];
  while (f!=0)
  {
    i++;
    exponents[i,1..nvars(basering)]=leadexp(f);
    f=f-lead(f);
  }
  if (size(#)==0)
  {
    #[1]="newtonPolytope";
  }
  // call polymakePolytope with exponents
  return(polymakePolytope(exponents,#));
}
example
{
   "EXAMPLE:";
   echo=2;
   ring r=0,(x,y,z),dp;
   matrix M[4][1]=1,x,y,z;
   poly f=y3+x2+xy+2xz+yz+z2+1;
   // the Newton polytope of f is
   list np=newtonPolytope(f);
   // the vertices of the Newton polytope are:
   np[1];
   // its dimension is
   np[2];
   // np[3] contains information how the vertices are connected to each other,
   // e.g. the first vertex (number 0) is connected to the second, third and
   //      fourth vertex
   np[3][1];
   //////////////////////////
   f=x2-y3;
   // the Newton polytope of f is
   np=newtonPolytope(f);
   // the vertices of the Newton polytope are:
   np[1];
   // its dimension is
   np[2];
   // the Newton polytope is contained in the affine space given
   //     by the equations
   np[4]*M;
}

proc newtonPolytopeLP (poly f)
"USAGE:  newtonPolytopeLP(f);  f poly
RETURN: list, the exponent vectors of the monomials occuring in f, i.e. the lattice
              points of the Newton polytope of f
EXAMPLE: example normalFan;   shows an example"
{
  list np;
  int i=1;
  while (f!=0)
  {
    np[i]=leadexp(f);
    f=f-lead(f);
    i++;
  }
  return(np);
}
example
{
   "EXAMPLE:";
   echo=2;
   ring r=0,(x,y,z),dp;
   poly f=y3+x2+xy+2xz+yz+z2+1;
   // the lattice points of the Newton polytope of f are
   newtonPolytopeLP(f);
}

proc normalFan (intmat vertices,intmat affinehull,list graph,int er,list #)
"USAGE:  normalFan (vert,aff,graph,rays,[,#]);   vert,aff intmat,  graph list, rays int, # string
ASSUME:  - vert is an integer matrix whose rows are the coordinate of the vertices of
           a convex lattice polytope; 
@*       - aff describes the affine hull of this polytope, i.e.
           the smallest affine space containing it, in the following sense:
           denote by n the number of columns of vert, then multiply aff by (1,x(1),...,x(n))
           and set the resulting terms to zero in order to get the equations for the affine hull;
@*       - the ith entry of graph is an integer vector describing to which vertices
           the ith vertex is connected, i.e. a k as entry means that the vertex vert[i] is
           connected to vert[k];
@*       - the integer rays is either one (if the extreme rays should be computed) or zero (otherwise)
RETURN:  list, the ith entry of L[1] contains information about the cone in the normal fan
               dual to the ith vertex of the polytope
@*             L[1][i][1] = integer matrix representing the inequalities which describe the
                            cone dual to the ith vertex
@*             L[1][i][2] = a list which contains the inequalities represented by L[i][1]
                            as a list of strings, where we use the variables x(1),...,x(n)
@*             L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix
                            whose rows are the extreme rays of the cone
@*             L[2] = is an integer matrix whose rows span the linearity space of the fan,
                      i.e. the linear space which is contained in each cone
NOTE:    - the procedure calls for its computation polymake by Ewgenij Gawrilow,
           TU Berlin and Michael Joswig, so it only works if polymake is installed;
           see http://www.math.tu-berlin.de/polymake/
@*       - in the optional argument # it is possible to hand over other names for the
           variables to be used -- be carful, the format must be correct and that is
           not tested, e.g. if you want the variable names to be u00,u10,u01,u11
           then you must hand over the string u11,u10,u01,u11
EXAMPLE: example normalFan;   shows an example"
{
  list ineq; // stores the inequalities of the cones
  int i,j,k;
  // we work over the following ring
  execute("ring ineqring=0,x(1.."+string(ncols(vertices))+"),lp;");
  string greatersign=">";
  // create the variable names
  if (size(#)>0)
  {
    if (typeof(#[1])=="string")
    {
      kill ineqring;
      execute("ring ineqring=0,("+#[1]+"),lp;");
    }
    if (size(#)>1)
    {
      greatersign="<";
    }
  }
  //////////////////////////////////////////////////////////////////
  // Compute first the inequalities of the cones
  //////////////////////////////////////////////////////////////////
  matrix VAR[1][ncols(vertices)]=maxideal(1);
  matrix EXP[ncols(vertices)][1];
  poly p,pl,pr;
  // consider all vertices of the polytope
  for (i=1;i<=nrows(vertices);i++)
  {
    // first we produce for each vertex in the polytope
    // the inequalities describing the dual cone in the normal fan
    list pp;  // contain strings representing the inequalities describing the normal cone
    intmat ie[size(graph[i])][ncols(vertices)]; // contains the inequalities as rows
    // consider all the vertices to which the ith vertex in the polytope is connected by an edge
    for (j=1;j<=size(graph[i]);j++)
    {
      // produce the vector ie_j pointing from the jth vertex to the ith vertex;
      // this will be the jth inequality for the cone in the normal fan dual to the ith vertex
      ie[j,1..ncols(vertices)]=vertices[i,1..ncols(vertices)]-vertices[graph[i][j],1..ncols(vertices)];
      EXP=ie[j,1..ncols(vertices)];
      // build a linear polynomial with the entries of ie_j as coefficients
      p=(VAR*EXP)[1,1];
      pl,pr=0,0;
      // separate the terms with positive coefficients in p from those with negative coefficients
      for (k=1;k<=size(p);k++)
      {
        if (leadcoef(p[k])<0)
        {
          pr=pr-p[k];
        }
        else
        {
          pl=pl+p[k];
        }
      }
      // build the string which represents the jth inequality for the cone dual to the ith vertex
      // as polynomial inequality of type string, and store this in the list pp as jth entry
      pp[j]=string(pl)+" "+greatersign+" "+string(pr);
    }
    // all inequalities for the ith vertex are stored in the list ineq
    ineq[i]=list(ie,pp);
    kill ie,pp; // kill certain lists
  }
  // remove the first column of affine hull to compute the linearity space
  intmat linearity=intmatcoldelete(affinehull,1);
  //////////////////////////////////////////////////////////////////
  // Compute next the extreme rays of the cones
  //////////////////////////////////////////////////////////////////
  if (er==1)
  {
    list extremerays; // keeps the result
    string polymake; // keeps polymake output
    // the header for ineq.polymake
    string head="_application polytope
_version 2.2
_type RationalPolytope

INEQUALITIES
";
    // the tail for both polymake files
    string tail="
EQUATIONS
";
    tail=tail+intmatToPolymake(linearity,"rays");
    string ungleichungen; // keeps the inequalities for the polymake code
    intmat M; // the matrix keeping the inequalities
    // create the file ineq.output
    write(":w /tmp/ineq.output","");
    int dimension; // keeps the dimension of the intersection the bad cones with the u11tobeseencone
    for (i=1;i<=size(ineq);i++)
    {
      i,". Cone of ",nrows(vertices); // indicate how many vertices have been dealt with
      ungleichungen=intmatToPolymake(ineq[i][1],"rays");
      // write the inequalities to ineq.polymake and call polymake
      write(":w /tmp/ineq.polymake",head+ungleichungen+tail);
      ungleichungen=""; // clear ungleichungen
      system("sh","cd /tmp; /bin/rm ineq.output; polymake ineq.polymake VERTICES > ineq.output");
      // read the result of polymake
      polymake=read("/tmp/ineq.output");
      intmat VERT=polymakeToIntmat(polymake,"affine");
      extremerays[i]=VERT;
      kill VERT;
    }
    for (i=1;i<=size(ineq);i++)
    {
      ineq[i]=ineq[i]+list(extremerays[i]);
    }
  }
  // delete the tmp-files, if polymakekeeptmpfiles is not set
  if (defined(polymakekeeptmpfiles)==0)
  {
    system("sh","/bin/rm /tmp/ineq.polymake");
    system("sh","/bin/rm /tmp/ineq.output");
  }
  // get the linearity space
  return(list(ineq,linearity));
}
example
{
   "EXAMPLE:";
   echo=2;
   ring r=0,(x,y,z),dp;
   matrix M[4][1]=1,x,y,z;
   poly f=y3+x2+xy+2xz+yz+z2+1;
   // the Newton polytope of f is
   list np=newtonPolytope(f);
   // the Groebner fan of f, i.e. the normal fan of the Newton polytope
   list gf=normalFan(np[1],np[4],np[3],1,"x,y,z");
   // the number of cones in the Groebner fan of f is:
   size(gf[1]);
   // the inequalities of the first cone as matrix are:
   print(gf[1][1][1]);
   // the inequalities of the first cone as string are:
   print(gf[1][1][2]);
   // the rows of the following matrix are the extreme rays of the first cone:
   print(gf[1][1][3]);
   // each cone contains the linearity space spanned by:
   print(gf[2]);
}

proc groebnerFan (poly f,list #)
"USAGE:  groebnerFan(f[,#]);  f poly, # string
RETURN:  list, the ith entry of L[1] contains information about the ith cone in the Groebner fan
               dual to the ith vertex in the Newton polytope of the f
@*             L[1][i][1] = integer matrix representing the inequalities which describe the cone
@*             L[1][i][2] = a list which contains the inequalities represented by L[1][i][1]
                            as a list of strings
@*             L[1][i][3] = an interger matrix whose rows are the extreme rays of the cone
@*             L[2] = is an integer matrix whose rows span the linearity space of the fan,
                      i.e. the linear space which is contained in each cone
@*             L[3] = the Newton polytope of f in the format of the procedure newtonPolytope
@*             L[4] = integer matrix where each row represents the exponet vector of one monomial
                      occuring in the input polynomial
NOTE:    - if you have alread computed the Newton polytope of f then you might want
           to use the procedure normalFan instead in order to avoid doing costly computation
           twice
@*       - the procedure calls for its computation polymake by Ewgenij Gawrilow,
           TU Berlin and Michael Joswig, so it only works if polymake is installed;
           see http://www.math.tu-berlin.de/polymake/
@*       - the procedure creates the file  /tmp/newtonPolytope.polymake which contains the
           Newton polytope of f in polymake format and which can be used for further
           computations with polymake
@*       - it is possible to give as an optional second argument as string which then will be
           used instead of 'newtonPolytope' in the name of the polymake output file
EXAMPLE: example groebnerFan;   shows an example"
{
  int i,j;
  // compute the list of exponent vectors of the polynomial, which are the lattice points
  // whose convex hull is the Newton polytope of f
  intmat exponents[size(f)][nvars(basering)];
  while (f!=0)
  {
    i++;
    exponents[i,1..nvars(basering)]=leadexp(f);
    f=f-lead(f);
  }
  if (size(#)==0)
  {
    #[1]="newtonPolytope";
  }
  // call polymakePolytope with exponents
  list newtonp=polymakePolytope(exponents,"newtonPolytope");
  // get the variables as string
  string variablen;
  for (i=1;i<=nvars(basering);i++)
  {
    variablen=variablen+string(var(i))+",";
  }
  variablen=variablen[1,size(variablen)-1];
  // call normalFan in order to compute the Groebner fan
  list gf=normalFan(newtonp[1],newtonp[4],newtonp[3],1,variablen);
  // append newtonp to gf
  gf[3]=newtonp;
  // append the exponent vectors to gf
  gf[4]=exponents;
  return(gf);
}
example
{
   "EXAMPLE:";
   echo=2;
   ring r=0,(x,y,z),dp;
   matrix M[4][1]=1,x,y,z;
   poly f=y3+x2+xy+2xz+yz+z2+1;
   // the Newton polytope of f is
   list gf=groebnerFan(f);
   // the exponent vectors of f are ordered as follows
   gf[4];
   // the first cone of the groebner fan has the inequalities
   gf[1][1][1];
   // as a string they look like
   gf[1][1][2];
   // and it has the extreme rays
   print(gf[1][1][3]);
   // the linearity space is spanned by
   print(gf[2]);
   // the vertices of the Newton polytope are:
   gf[3][1];
   // its dimension is
   gf[3][2];
   // np[3] contains information how the vertices are connected to each other,
   // e.g. the 1st vertex is connected to the 2nd, 3rd and 4th vertex
   gf[3][3][1];
}



proc intmatToPolymake (intmat M,string art)
"USAGE:  intmatToPolymake(M,art);  M intmat, art string
ASSUME:  - M is an integer matrix which should be transformed into polymake format;
@*       - 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:  string, the matrix is transformed in a string and a first column has been added
EXAMPLE: example intmatToPolymake;   shows an example"
{
  if (art=="rays")
  {
    string anf="0 ";
  }
  else
  {
    string anf="1 ";
  }
  string sp;
  int i,j;
  // create the lattice point list for polymake
  for (i=1;i<=nrows(M);i++)
  {
    sp=sp+anf;
    for (j=1;j<=ncols(M);j++)
    {
      sp=sp+string(M[i,j])+" ";
      if (j==ncols(M))
      {
        sp=sp+"
";
      }
    }
  }
  return(sp);
}
example
{
   "EXAMPLE:";
   echo=2;
   intmat M[3][4]=1,2,3,4,5,6,7,8,9,10,11,12;
   intmatToPolymake(M,"rays");
   intmatToPolymake(M,"points");
}

proc polymakeToIntmat (string pm,string art)
"USAGE:  polymakeToIntmat(pm,art);  pm, art string
ASSUME:  pm is the result of calling polymake with one 'argument' like VERTICES, AFFINE_HULL, etc.,
         so that the first row of the string is the name of the corresponding 'argument' and
         the further rows contain the result which consists of vectors either over the integers
         or over the rationals
RETURN:  intmat, the rows of the matrix are basically the vectors in pm from the second row on
                 where each row has been multiplied with the lowest common multiple of the
                 denominators of its entries so as to be an integer matrix; moreover,
                 if art=='affine', then the first column is omitted since we only want affine
                 coordinates
EXAMPLE: example polymakeToIntmat;   shows an example"
{
  // we need a line break
  string zeilenumbruch="
";
  // remove the 'argment' name, i.e. the first row of pm
  while (pm[1]!=zeilenumbruch)
  {
    pm=stringdelete(pm,1);
  }
  pm=stringdelete(pm,1);
  // find out how many entries each vector has, namely one more than 'spaces' in a row
  int i=1;
  int s=1;
  int z=1;
  while (pm[i]!=zeilenumbruch)
  {
    if (pm[i]==" ")
    {
      s++;
    }
    i++;
  }
  // if we want to have affine coordinates
  if (art=="affine")
  {
    s--; // then there is one column less
    // and the entry of the first column (in the first row) has to be removed
    while (pm[1]!=" ")
    {
      pm=stringdelete(pm,1);
    }
    pm=stringdelete(pm,1);
  }
  // we add two line breaks at the end in order to have this as a stopping criterion
  pm=pm+zeilenumbruch+zeilenumbruch;
  // we now have to work through each row
  for (i=1;i<=size(pm);i++)
  {
    // if there are two consecutive line breaks we are done
    if ((pm[i]==zeilenumbruch) and (pm[i+1]==zeilenumbruch))
    {
      i=size(pm)+1;
    }
    else
    {
      // a line break has to be replaced by a comma
      if (pm[i]==zeilenumbruch)
      {
        z++;
        pm[i]=",";
        // if we want to have affine coordinates, then we have to delete the first entry in each row
        if (art=="affine")
        {
         while (pm[i+1]!=" ")
         {
           pm=stringdelete(pm,i+1);
         }
         pm=stringdelete(pm,i+1);
        }
      }
      // a space has to be replaced by a comma
      if (pm[i]==" ")
      {
        pm[i]=",";
      }
    }
  }
  // if we have introduced superflous commata at the end, they should be removed
  while (pm[size(pm)]==",")
  {
    pm=stringdelete(pm,size(pm));
  }
  // since the matrix could be over the rationals, we need a ring with rational coefficients
  ring zwischering=0,x,lp;
  // create the matrix with the elements of pm as entries
  execute("matrix mm["+string(z)+"]["+string(s)+"]="+pm+";");
  // transform this into an integer matrix
  matrix M[1][ncols(mm)]; // takes a row of mm
  int cm;  // takes a lowest common multiple
  // multiply each row by an integer such that its entries are integers
  for (int j=1;j<=nrows(mm);j++)
  {
    M=mm[j,1..ncols(mm)];
    cm=commondenominator(M);
    for (i=1;i<=ncols(mm);i++)
    {
      mm[j,i]=cm*mm[j,i];
    }
  }
  // transform the matrix mm into an integer matrix
  execute("intmat im["+string(z)+"]["+string(s)+"]="+string(mm)+";");
  return(im);
}
example
{
   "EXAMPLE:";
   echo=2;
   // this is the usual output of some polymake computation
   string pm="VERTICES
0 1 3 5/3 1/3 -1 -23/3 -1/3 5/3 1/3 1
0 1 3 -23/3 5/3 1 5/3 1/3 1/3 -1/3 -1
0 1 1 1/3 -1/3 -1 5/3 1/3 -23/3 5/3 3
0 1 1 5/3 -23/3 3 1/3 5/3 -1/3 1/3 -1
0 1 -1 1/3 5/3 3 -1/3 -23/3 1/3 5/3 1
0 1 -1 -1/3 1/3 1 1/3 5/3 5/3 -23/3 3
0 1 -1 1 3 -5 -1 3 -1 1 -1
0 1 -1 -1 -1 -1 1 1 3 3 -5
0 1 -5 3 1 -1 3 -1 1 -1 -1

";
   intmat PM=polymakeToIntmat(pm,"affine");
   // note that the first column has been removed, since we asked for
   // affine coordinates, and the denominators have been cleared
   print(PM);
}

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

proc triangulations (list polygon)
"USAGE:  triangulations(polygon); list polygon
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.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM
@*     - 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 use the procedure polymakeKeepTmpFiles in before
@*     - 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;
  // 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
  system("sh","cd /tmp; points2triangs < triangulationsinput > triangulationsoutput");
  string p2t=read("/tmp/triangulationsoutput"); // takes the result of points2triangs
  // delete the tmp-files, if polymakekeeptmpfiles is not set
  if (defined(polymakekeeptmpfiles)==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
            {
              np2t=np2t+p2t[i];
            }
          }
        }
      }
    }
  }
  list T;
  execute(np2t);
  // 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.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/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 USING POLYMAKE AND TOPCOM
////////////////////////////////////////////////////////////////////////////////////////

proc secondaryFan (list polygon,list #)
"USAGE:  secondaryFan(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, the ith entry of L[1] contains information about the ith cone in the
               secondary fan of the polygon, i.e. the cone dual to the ith vertex of the
               secondary polytope
@*             L[1][i][1] = integer matrix representing the inequalities which describe the
                            cone dual to the ith vertex
@*             L[1][i][2] = a list which contains the inequalities represented by L[1][i][1]
                            as a list of strings, where we use the variables x(1),...,x(n)
@*             L[1][i][3] = only present if 'er' is set to 1; in that case it is an interger matrix
                            whose rows are the extreme rays of the cone
@*             L[2] = is an integer matrix whose rows span the linearity space of the fan,
                      i.e. the linear space which is contained in each cone
@*             L[3] = the secondary polytope in the format of the procedure polymakePolytope
@*             L[4] = the list of triangulations corresponding to the vertices
                      of the secondary polytope
NOTE:    - the procedure calls for its computation polymake by Ewgenij Gawrilow,
           TU Berlin and Michael Joswig, so it only works if polymake is installed;
           see http://www.math.tu-berlin.de/polymake/
@*       - in the optional argument # it is possible to hand over other names for the
           variables to be used -- be carful, the format must be correct and that is
           not tested, e.g. if you want the variable names to be u00,u10,u01,u11
           then you must hand over the string u11,u10,u01,u11
@*       - 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.uni-bayreuth.de/departments/wirtschaftsmathematik/rambau/TOPCOM
EXAMPLE: example secondaryFan;   shows an example"
{
  if (size(#)==0)
  {
    list triang=triangulations(polygon);
  }
  else
  {
    list triang=#[1];
  }
  list sp=secondaryPolytope(polygon,triang);
  list spp=polymakePolytope(sp[1]);
  list sf=normalFan(spp[1],spp[4],spp[3],1);
  return(list(sf[1],sf[2],spp,triang));
}
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 secfan=secondaryFan(polygon);
   // the number of cones in the secondary fan of the polygon
   size(secfan[1]);
   // the inequalities of the first cone as matrix are:
   print(secfan[1][1][1]);
   // the inequalities of the first cone as string are:
   print(secfan[1][1][2]);
   // the rows of the following matrix are the extreme rays of the first cone:
   print(secfan[1][1][3]);
   // each cone contains the linearity space spanned by:
   print(secfan[2]);
   // the points in the secondary polytope
   print(secfan[3][1]);
   // the corresponding triangulations
   secfan[4];
}


////////////////////////////////////////////////////////////////////////////////////////
/// 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="ring cyclering=0,(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+"),lp;";
  execute(rst);
  // 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 the 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, the coordinates of the 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 build 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 polygone 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 either 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 followin 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];  // the 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
  // ly 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 can not 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;
      minimisedorderedvertices[size(minimisedorderedvertices)+1]=testvertex; // store the next corner of NSD
      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 ly 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)/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 the lattice lengths of the edges of the Newton cylce
  // 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 vertix 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=newtonPolytopeLP(f);
   // 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];
}


proc ellipticNFDB (int n,list #)
"USAGE:  ellipticNFDB(n[,#]);   n int, # list
ASSUME:  n is an intger between 1 and 16
PURPOSE: this is a database storing the 16 normal forms of planar polygons with
         precisely one interior point up to unimodular affine transformations
@*       (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 nth normal form
@*             L[2] : list whose entries are all the lattice points of the nth normal form
@*             L[3] : only present if the optional parameter # is present, and then
                      it is a polynomial in the variables (x,y) whose Newton polygon
                      is the nth normal form
NOTE:    the optional parameter is only allowed if the basering has the variables x and y
EXAMPLE: example ellipticNFDB;   shows an example"
{
  if ((n<1) or (n>16))
  {
    ERROR("n is not between 1 and 16.");
  }
  if (size(#)>0)
  {
    if ((defined(x)==0) or (defined(y)==0))
    {
      ERROR("The variables x and y are not defined.");
    }
  }
  if ((defined(x)==0) or (defined(y)==0))
  {
    ring nfring=0,(x,y),dp;
  }
  // store the normal forms as polynomials
  list nf=x2+y+xy2,x2+x+1+xy2,x3+x2+x+1+y2+y,x4+x3+x2+x+1+y2+y+x2y,x3+x2+x+1+x2y+y+xy2+y2+y3,
    x2+x+x2y+y2,x2y+x+y+xy2,x2+x+1+y+xy2,x2+x+1+y+xy2+y2,x3+x2+x+1+x2y+y+y2,x3+x2+x+1+x2y+y+xy2+y2,
    x2+x+1+x2y+y+x2y2+xy2+y2,x+1+x2y+y+xy2,x2+x+1+x2y+y+xy2,x2+x+1+x2y+y+xy2+y2,x2+x+x2y+y+xy2+y2;
  list pg=newtonPolytopeLP(nf[n]);
  if (size(#)==0)
  {
    return(list(findOrientedBoundary(pg)[2],pg));
  }
  else
  {
    return(list(findOrientedBoundary(pg)[2],pg,nf[n]));
  }
}
example
{
   "EXAMPLE:";
   echo=2;
   list nf=ellipticNFDB(5);
   // the vertices of the 5th normal form are
   nf[1];
   // its lattice points are
   nf[2];
}


/////////////////////////////////////////////////////////////////////////////////
/// AUXILARY PROCEDURES
/////////////////////////////////////////////////////////////////////////////////

proc polymakeKeepTmpFiles (int i)
"USAGE:  polymakeKeepTmpFiles(int i);   i int
PURPOSE: some procedures create files in the directory /tmp which are used for
         computations with polymake respectively topcom; these will be removed
         when the corresponding procedure is left; however, it might be desireable
         to keep them for further computations with either polymake or topcom; this
         can be achieved by this procedure; call the procedure as:
@*       - polymakeKeepTmpFiles(1);  - then the files will be kept
@*       - polymakeKeepTmpFiles(0);  - then the files will be removed in the future
RETURN:  none"
{
  if ((i==1) and (defined(polymakekeeptmpfiles)==0))
  {
    int polymakekeeptmpfiles;
    export(polymakekeeptmpfiles);
  }
  if (i!=1)
  {
    if (defined(polymakekeeptmpfiles))
    {
      kill polymakekeeptmpfiles;
    }
  }
}


/////////////////////////////////////////////////////////////////////////////////
/////////////////////////////////////////////////////////////////////////////////
/// AUXILARY 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 (intmat 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 and normalFan"
{
  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));
  }
}

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 seperated 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));
}