source: git/Singular/LIB/realizationMatroids.lib @ b0732eb

/////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Tropical Geometry";
info="
LIBRARY:  realizationMatroids.lib   Deciding Relative Realizability for Tropical Fan Curves in 2-Dimensional Matroidal Fans
AUTHORS:  Anna Lena Winstel, winstel@mathematik.uni-kl.de
OVERVIEW: In tropical geometry, one question to ask is the following: given a one-dimensional balanced polyhedral fan C which is set theoretically contained in the tropicalization trop(Y) of an algebraic variety Y, does there exist a curve X in Y such that trop(X) = C? This equality of C and trop(X) denotes an equality of both, the fans trop(X) and C and their weights on the maximal cones. The relative realization space of C with respect to Y is the space of all algebraic curves in Y which tropicalize to C.

This library provides procedures deciding relative realizability for tropical fan curves, i.e. one-dimensional weighted balanced polyhedral fans, contained in two-dimensional matroidal fans trop(Y) where Y is a projective plane.

NOTATION: If Y is a projective plane in (n-1)-dimensional projective space, we consider trop(Y) in R^n/<1>. Moreover, for the relative realization space of C with respect to Y we only consider algebraic curves of degree deg(C) in Y which tropicalize to C.

PROCEDURES:

realizationDim(I,C);          For a given tropical fan curve C in trop(Y), where Y = V(I) is a projective plane, this routine returns the dimension of the relative realization space of C with respect to Y, that is the space of all algebraic curves of degree deg(C) in Y which tropicalize to C. If the realization space is empty, the output is set to -1.

irrRealizationDim(I,C);       This routine returns the dimension of the irreducible relative realization space of the tropical fan curve C with respect to Y = V(I), that is the space of all irreducible algebraic curves of degree deg(C) in Y which tropicalize to C. If the irreducible relative realization space is empty, the output is set to -1.

realizationDimPoly(I,C);      If C is a tropical fan curve contained in the tropicalization trop(Y) of the projective plane Y = V(I) such that the relative realization space M of C is non-empty, this routine returns the tuple (dim(M),f) where f is an example of a homogeneous polynomial of degree deg(C) cutting out a curve X in Y which tropicalizes to C. If M is empty, the output is set to -1.

";

LIB "control.lib";
LIB "qhmoduli.lib";

static proc gcdvector(intvec v)
{
  int i;
  int ggt = 0;
  for(i=1;i<=size(v);i++)
  {
    ggt = gcd(ggt,v[i]);
    if( ggt == 1 )
    {
      return(ggt);
    }
  }
  return(ggt);
}

static proc balanced(list lInput)
{
  list ba;
  int i;
  int j;
  if(size(lInput)>0)
  {
    for(i=1;i<=size(lInput[1]);i++) 
    {
      ba[i] = 0;
      for(j=1;j<=size(lInput);j++)
      {
        ba[i] = ba[i] + lInput[j][i];
      }
    }
    int boolean = 1;
    for(i=2;i<=size(ba);i++)
    {
      if(ba[i] != ba[1])
      {
        boolean = 0;
      }
    }
    if(boolean == 1)
    {
      return(ba[1]);
    }
    else
    {
      return(0);
    }
  }
  else
  {
    return(0);
  }
}

static proc genPoly(int d, int i, int j, int k)
{
  int ii;
  int ij;
  int ik = 1;
  poly f = 0;
  for(ii=0;ii<=d;ii++)
  {
    for(ij=0;ij<=d-ii;ij++)
    {
       f = f + a(ik)*x(i)^(d-ii-ij)*x(j)^ij*x(k)^ii;
       ik = ik + 1;
    }
  }
  return(f);
}

static proc prodvar(int n)
{
  int i;
  poly f = 1;
  for(i=1;i<=n;i++)
  {
    f = f * x(i);
  }
  return(f);
}

static proc lessThan(int i, int j, intvec v, intvec w)
{
  number a = v[i];
  number b = v[j];
  number c = w[i];
  number d = w[j];
  if((a/b)<(c/d))
  { return(1); }
  else
  { return(0); }
}

static proc sortSlope(int i, int j, list lInput)
{
  int k;
  int l;
  intvec v;
  for(k=1;k<size(lInput);k++)
  {
    for(l=1;l<=size(lInput)-k;l++)
    {
      if(lessThan(i,j,lInput[l+1],lInput[l]))
      {
        v = lInput[l];
        lInput[l] = lInput[l+1];
        lInput[l+1] = v;
      }
    }
  }
  return(lInput);
}

static proc coefMonomial(poly f, poly g, int n)
{
  matrix m = coef(f,prodvar(n));
  poly h;
  for(int i=1;i<=ncols(m);i++)
  {
    if(m[1,i] == g)
    {
      h = m[2,i];
    }
  }
  return(h);
}

static proc ismultiple(intvec v, intvec w)
{
  int boolean = 1;
  if(v[1] != 0)
  {
    if((number(w[2]) == number(v[2])*number(w[1])/number(v[1])) and (number(w[3]) == number(v[3])*number(w[1])/number(v[1])))
    {
      return(1);
    }
    else 
    {
      return(0);
    }
  }
  else
  {
    if(v[2] != 0)
    {
      if((number(w[1]) == number(v[1])*number(w[2])/number(v[2])) and (number(w[3]) == number(v[3])*number(w[2])/number(v[2])))
      {
        return(1);
      }
      else 
      {
        return(0);
      }
    }
    else
    {
      if((w[2] == 0) and (w[1] == 0))
      {
        return(1);
      }
      else
      {
        return(0);
      }
    }
  }
}

static proc simplifyList(list lInput);
{
  int i;
  int k = size(lInput);
  for(i=1;i<=k;i++)
  {
    if(lInput[i] == intvec(0,0,0))
    {
      lInput = delete(lInput,i);
      k = k-1;
      i = i-1;
    }
  }
  k = size(lInput);
  int j;
  for(i=1;i<k;i++)
  {
    for(j=i+1;j<=k;j++)
    {
      if(ismultiple(lInput[i],lInput[j]))
      {
        lInput[i] = lInput[i] + lInput[j];
        lInput = delete(lInput,j);
        j = j-1;
        k = k-1;
      }
    }
  }
  return(lInput);
}

static proc realizationDimIdeal(ideal iInput, list lInput)
{
  //normalize the vectors
  intvec helpintvec = 1;
  int i;
  int c;
  for(i=1;i<size(lInput[1]);i++)
  {
    helpintvec = helpintvec,1;
  }
  for(i=1;i<=size(lInput);i++)
  {
    lInput[i] = lInput[i] - Min(lInput[i])*helpintvec;
  }

  //check if the curve is balanced and compute its degree
  int d = balanced(lInput);
  if(d == 0)
  {
    printf("The curve is not balanced.");
    return(-2);
  }

  //change basering, store the actual basering in a variable
  def save = basering;
  int n = size(lInput[1]);
  int N = (d+2)*(d+1) div 2;
  ring r1;     
  ring r = 0,(x(1..n),a(1..N),t),dp;
  setring r;
  ideal I = fetch(save,iInput);
  I = std(I);

  if(dim(I) != (4+N))
  {
    printf("The ideal is not defining a projective plane.");
    return(-2);
  }

  //for any three variables, compute the projection
  int i2;
  int i3;
  int i4;
  int j;
  int k;
  int l;
  int i_w;
  int good;
  int i_good = 0;
  list P;
  intvec v;
  intvec w;
  ideal E;
  list NE;
  list S1;
  list S2;
  list S3;
  poly h;
  poly g;
  poly coefMon;
  matrix F;
  matrix G;
  list listunitvec;
  v = 1;
  for(i=2;i<=n+N+1;i++)
  {
    v = v,0;
  }
  listunitvec = list(v);
  for(i=2;i<=n;i++)
  {
    v = 0;
    for(j=2;j<=n+N+1;j++)
    {
      if(i != j)
      {
        v = v,0;
      }
      else
      {
        v = v,1;
      }
    }
    listunitvec = listunitvec + list(v);
  }
  intmat M[n+N+1][n+N+1];
  for(i=n+1;i<=n+N+1;i++)
  {
    M[i,i] = 1;
  }
  int i_start;
  list luv1;
  intmat M1[n+N+1][n+N+1];
  for(i=1;i<=n-2;i++)
  {
    for(j=i+1;j<=n-1;j++)
    {
      for(k=j+1;k<=n;k++)
      {
        //compute the algebraic projection
        luv1 = listunitvec;
        luv1 = delete(luv1,k);
        luv1 = delete(luv1,j);
        luv1 = delete(luv1,i);
        luv1 = luv1 + list(listunitvec[i],listunitvec[j],listunitvec[k]);
        M1 = M;
        for(l=1;l<=size(luv1);l++)
        {
          M1[l,1..(n+N+1)] = luv1[l];
        }
        r1 = ring(0,(x(1..n),a(1..N),t),M(M1));
        setring r1;
        ideal Ir1 = fetch(r,I);
        Ir1 = std(Ir1);
        //check if this projection is "good"
        good = 1;
        ideal Ii = x(i);
        ideal Ij = x(j);
        ideal Ik = x(k);
        for(l=1;l<=size(Ir1);l++)
        {
          if((reduce(lead(Ir1[l]),Ii) == 0) or (reduce(lead(Ir1[l]),Ij) == 0) or (reduce(lead(Ir1[l]),Ik) == 0))
          {
            good = 0;
          }
        }
        if(good == 1)
        {
          //for the first "good" projection, initialise the general polynomial f
          if(i_good == 0)
          {
            setring r;
            poly f = genPoly(d,i,j,k);
            setring r1;
            i_good = 1;
            intvec vgood = i,j,k;
          }
          poly fr1 = fetch(r,f);
          poly hr1 = reduce(fr1,Ir1);
          setring r;
          h = fetch(r1,hr1);
          //compute the tropical projection
          P = list();
          for(l=1;l<=size(lInput);l++)
          {
            v = lInput[l][i],lInput[l][j],lInput[l][k];
            P[l] = v;
          }
          P = simplifyList(P);
          //collect the conditions coming from the Newton polytopes
          S1 = list();
          S2 = list();
          S3 = list();
          for(l=1;l<=size(P);l++)
          {
            if((P[l][1] != 0) and (P[l][2] != 0))
            { S3 = S3 + list(P[l]); }
            if((P[l][2] != 0) and (P[l][3] != 0))
            { S1 = S1 + list(P[l]); }
            if((P[l][1] != 0) and (P[l][3] != 0))
            { S2 = S2 + list(P[l]); }
          }
          //sort the lists
          S1 = sortSlope(3,2,S1);
          S2 = sortSlope(1,3,S2);
          S3 = sortSlope(2,1,S3);
          //find conditions from S1
          i_start = 0;
          for(l=1;l<=size(S1);l++)
          {
            i_start = i_start + S1[l][2];
          }
          //find starting point
          w = intvec(0,i_start);
          coefMon = coefMonomial(h,x(k)^(w[2])*x(i)^(d-w[2]),n);
          NE = NE + list(coefMon);
          for(i2=1;i2<=size(S1);i2++)
          {
            w[1] = w[1] + S1[i2][3];
            w[2] = w[2] - S1[i2][2];
            coefMon = coefMonomial(h,x(k)^(w[2])*x(j)^(w[1])*x(i)^(d-w[1]-w[2]),n);
            NE = NE + list(coefMon);
            g = subst(h,x(j),x(j)*t^(S1[i2][2]),x(k),x(k)*t^(S1[i2][3]));
            i_w = S1[i2][2]*w[1] + S1[i2][3]*w[2];
            F = coeffs(g,t);
            for(i3=1;i3<=i_w;i3++)
            {
              G = coef(F[i3,1],prodvar(n));
              for(i4=1;i4<=ncols(G);i4++)
              {
                      E = E + ideal(G[2,i4]);
              }
            }
          }
          //find conditions from S2
          i_start = 0;
          for(l=1;l<=size(S2);l++)
          {
            i_start = i_start + S2[l][3];
          }
          //find starting point
          w = intvec(i_start,0);
          coefMon = coefMonomial(h,x(i)^(w[1])*x(j)^(d-w[1]),n);
          NE = NE + list(coefMon);
          for(i2=1;i2<=size(S2);i2++)
          {
            w[1] = w[1] - S2[i2][3];
            w[2] = w[2] + S2[i2][1];
            coefMon = coefMonomial(h,x(i)^(w[1])*x(k)^(w[2])*x(j)^(d-w[1]-w[2]),n);
            NE = NE + list(coefMon);
            g = subst(h,x(i),x(i)*t^(S2[i2][1]),x(k),x(k)*t^(S2[i2][3]));
            i_w = S2[i2][3]*w[2] + S2[i2][1]*w[1];
            F = coeffs(g,t);
            for(i3=1;i3<=i_w;i3++)
            {
              G = coef(F[i3,1],prodvar(n));
              for(i4=1;i4<=ncols(G);i4++)
              {
                      E = E + ideal(G[2,i4]);
              }
            }
          }
          //find conditions from S3
          i_start = 0;
          for(l=1;l<=size(S3);l++)
          {
            i_start = i_start + S3[l][1];
          }
          //find starting point
          w = intvec(0,i_start);
          coefMon = coefMonomial(h,x(j)^(w[2])*x(k)^(d-w[2]),n);
          NE = NE + list(coefMon);
          for(i2=1;i2<=size(S3);i2++)
          {
            w[1] = w[1] + S3[i2][2];
            w[2] = w[2] - S3[i2][1];
            coefMon = coefMonomial(h,x(i)^(w[1])*x(j)^(w[2])*x(k)^(d-w[1]-w[2]),n);
            NE = NE + list(coefMon);
            g = subst(h,x(i),x(i)*t^(S3[i2][1]),x(j),x(j)*t^(S3[i2][2]));
            i_w = S3[i2][2]*w[2] + S3[i2][1]*w[1];
            F = coeffs(g,t);
            for(i3=1;i3<=i_w;i3++)
            {
              G = coef(F[i3,1],prodvar(n));
              for(i4=1;i4<=ncols(G);i4++)
              {
                      E = E + ideal(G[2,i4]);
              }
            }
          }
        }
      }
    }
  }

  //check whether or not there is a common solution
  setring r;
  int isRealizable = 1;
  E = std(E);
  int i_dim = dim(E)-n-2;
  for(i=1;i<=size(NE);i++)
  { 
    if(reduce(NE[i],E) == 0)
    {
      isRealizable = 0;
    }
  }

  if(isRealizable == 1)
  {
    setring save;
    return(i_dim,fetch(r,E),fetch(r,NE),vgood);
  }
  else
  {
    return(-1);
  }
}

proc realizationDim(ideal iInput, list lInput)
"USAGE:   realizationDim(I,C); where I is a homogeneous linear ideal defining the projective plane Y = V(I) and C is a list of intvectors such that each intvector represents a one-dimensional cone in the tropical fan curve whose relative realizability should be checked. This representation is done in the following way: the one-dimensional cone K is represented by a vector w whose equivalence class [w] in R^n/<1> can be written as [w] = m*[v] where [v] is the primitive generator of K and m is the weight of K.
RETURNS:  the dimension of the relative realization space of the tropical curve C with respect to Y, and -1 if the relative realization space is empty.
EXAMPLE:  realizationDim; shows an example"
{
  int ret = realizationDimIdeal(iInput,lInput)[1];
  if(ret[1] == -2)
  {
    printf("WARNING: no computation possible, return value is not meaningful!");
    return(-2);
  }
  else
  {
    return(ret);
  }
}
example
{
  "EXAMPLE:"; echo=2;
  ring r = 0,(x(1..4)),dp;
  ideal I = x(1)+x(2)+x(3)+x(4);
  list C = list(intvec(2,2,0,0),intvec(0,0,2,1),intvec(0,0,0,1));
  //C represents the tropical fan curve which consists of the cones
  //cone([(1,1,0,0)]) (with weight 2), cone([(0,0,2,1)]) (with weight 1) 
  //and cone([(0,0,0,1)]) (with weight 1)
  realizationDim(I,C);
}

proc irrRealizationDim(ideal iInput, list lInput)
"USAGE:   irrRealizationDim(I,C); where I is a homogeneous linear ideal defining the projective plane Y = V(I) and C is a list of intvectors such that each intvector represents a one-dimensional cone in the tropical fan curve whose irreducible relative realizability should be checked. This representation is done in the following way: a one-dimensional cone K is represented by a vector w whose equivalence class [w] in R^n/<1> can be written as [w] = m*[v] where [v] is the primitive generator of K and m is the weight of K.
RETURNS:  the dimension of the irreducible relative realization space of C with respect to Y, and -1 if the irreducible realization space is empty.
EXAMPLE:  irrRealizationDim; shows an example"
{
  int i;
  int i_dim = realizationDim(iInput,lInput);
  if(i_dim > -1)
  {
    //check if also realizable by an irreducible curve
    list lweight;
    int i_rdim = -1;
    //substitute the vectors by a primitve one and store the multiplicities
    for(i=1;i<=size(lInput);i++)
    {
      lweight[i] = gcdvector(lInput[i]);
      lInput[i] = lInput[i] div lweight[i];
    }
    //find all decompositions into two tropical curves
    intvec tm;
    for(i=1;i<=size(lInput);i++)
    {
      tm[i] = 0;
    }
    int na;
    list C1;
    list C2;
    int dimC1;
    int dimC2;
    while(na==0)
    {
      na = 1;
      for(i=1;i<=size(lInput);i++)
      {
        if(tm[i] < lweight[i])
        {
          tm[i] = tm[i]+1;
          na = 0;
          i = size(lInput);
        }
        else
        {
          tm[i] = 0;
        }
      }
      if(na == 0)
      {
        C1 = list();
        C2 = list();
        for(i=1;i<=size(lInput);i++)
        {
          if(tm[i] > 0)
          {
            C1 = C1 + list(tm[i]*lInput[i]);
            if(tm[i] < lweight[i])
            {
              C2 = C2 + list((lweight[i]-tm[i])*lInput[i]);
            }
          }
          else
          {
            C2 = C2 + list(lweight[i]*lInput[i]);
          }
        }
        if((balanced(C2) != 0) and (balanced(C2) <= balanced(C1)))
        {
          dimC1 = realizationDim(iInput,C1);
          dimC2 = realizationDim(iInput,C2);
          if((dimC1 >= 0) and (dimC2 >= 0))
          {
            i_rdim = Max(intvec(i_rdim,dimC1 + dimC2));
          }
        }
      }
    }
    if(i_rdim < i_dim)
    {
      return(i_dim);
    }
    else
    {
      return(-1);
    }
  }
  else
  {
    return(-1);
  }
}
example
{
  "EXAMPLE:"; echo=2;
  ring r = 0,(x(1..4)),dp;
  ideal I = x(1)+x(2)+x(3)+x(4);
  list C = list(intvec(2,2,0,0),intvec(0,0,2,2));
  //C represents the tropical fan curve which consists of the cones 
  //cone([(1,1,0,0)]) and cone([(1,1,0,0)]), both with weight 2
  realizationDim(I,C);
  irrRealizationDim(I,C);
}

proc realizationDimPoly(ideal iInput, list lInput)
"USAGE:   realizationDimPoly(I,C); where I is a homogeneous linear ideal defining the projective plane Y = V(I) and C is a list of intvectors such that each intvector represents a one-dimensional cone in the tropical fan curve whose relative realizability should be checked. This representation is done in the following way: the one-dimensional cone K is represented by a vector w whose equivalence class [w] in R^n/<1> can be written as [w] = m*[v] where [v] is the primitive generator of K and m is the weight of K.
RETURNS:  If the relative realization space of the tropical fan curve C is non-empty, this routine returns the tuple (r,f), where r is the dimension of the relative realization space and f is an example of a homogeneous polynomial of degree deg(C) cutting out a curve X in Y which tropicalizes to C. In case the relative realization space is empty, the output is set to -1.
EXAMPLE:  realizationDimPoly; shows an example"
{
  def save = basering;
  int d = balanced(lInput);
  int n = size(lInput[1]);
  int N = (d+2)*(d+1) div 2;  
  int i; 
  ring r = 0,(x(1..n),a(1..N)),dp;
  list ret = realizationDimIdeal(fetch(save,iInput),lInput);
  int realdim = ret[1];
  if(realdim != -1)
  {
    ideal E = ret[2];
    list NE = ret[3];
    E = std(E);
    //find variables which are free to choose
    intvec v;
    for(i=1;i<=size(E);i++)
    {
      v = v + leadexp(E[i]);
    }
    int j;
    int k;
    int boolean;
    ideal E1;
    list NE1;
    poly f;
    poly g;
    //initialize the list of the free variables
    list lValues;
    if(size(v) > 1)
    {
      for(j=1;j<=N;j++)
      {
        if(v[j+n] == 0)
        {
          lValues = lValues + list(list(a(j),0));
        }
      }
    }
    else
    {
      for(j=1;j<=N;j++)
      {
        lValues = lValues + list(list(a(j),0));
      }
    }
    //try to find an easy solution
    boolean = 1;
    for(j=1;j<=size(lValues);j++)
    {
      if(boolean == 1)
      {
        lValues[j][2] = 0;
      }
      else
      {
        lValues[j][2] = lValues[j][2] + 1;
      }
      boolean = 1;
      E1 = E;
      for(i=1;i<=size(E1);i++)
      {
        for(k=1;k<=j;k++)
        {
          E1[i] = subst(E1[i],lValues[k][1],lValues[k][2]);
        }
      }
      NE1 = NE;
      for(i=1;i<=size(NE);i++)
      {
        for(k=1;k<=j;k++)
        {
          NE1[i] = subst(NE1[i],lValues[k][1],lValues[k][2]);
        }
      }
      E1 = std(E1);
      for(i=1;i<=size(NE1);i++)
      {
        if(reduce(NE1[i],E1) == 0)
        {
          boolean = 0;
        }
      }
      if(boolean == 0)
      {
        j = j-1;
      }
    }
    //compute the values of the dependent variables
    for(j=1;j<=size(E);j++)
    {
      f = E[j];
      for(k=1;k<=size(lValues);k++)
      {
        f = subst(f,lValues[k][1],lValues[k][2]);
      }
      if(leadcoef(f) != 1)
      {
        for(k=1;k<=size(lValues);k++)
        {
          lValues[k][2] = lValues[k][2] * leadcoef(f);
        }
      }
      f = subst(f,leadmonom(f),0);
      lValues = lValues + list(list(leadmonom(E[j]),-f));
    }
    g = genPoly(d,ret[4][1],ret[4][2],ret[4][3]);
    for(j=1;j<=N;j++)
    { 
      g = subst(g,lValues[j][1],lValues[j][2]);
    }
    setring save;
    return(realdim,fetch(r,g));
  }
  else
  {
    return(-1);
  }
}
example
{
  "EXAMPLE:"; echo=2;
  ring r = 0,(x(1..4)),dp;
  ideal I = x(1)+x(2)+x(3)+x(4);
  list C = list(intvec(2,2,0,0),intvec(0,0,2,2));
  //C represents the tropical fan curve which consists of the cones 
  //cone([(1,1,0,0)]) and cone([(1,1,0,0)]), both with weight 2
  realizationDimPoly(I,C);
  C = list(intvec(0,0,0,4),intvec(0,1,3,0),intvec(1,0,1,0),intvec(0,2,0,0),intvec(3,1,0,0));
  //C represents the tropical fan curve which consists of the cones
  //cone([(0,0,0,1)]) with weight 4,
  //cone([(0,1,3,0)]), cone([(1,0,1,0)]) both with weight 1,
  //cone([(0,1,0,0)]) with weight 2, and
  //cone([(3,1,0,0)]) with weight 1
  realizationDimPoly(I,C);
}