source: git/Singular/LIB/pointid.lib @ 393c47

////////////////////////////////////////////////////////////////////////////////
version="$Id$";
category="Commutative algebra";
info="
LIBRARY:  pointid.lib     Procedures for computing a factorized lex GB of
                          the vanishing ideal of a set of points via the
                          Axis-of-Evil Theorem (M.G. Marinari, T. Mora)

AUTHOR:   Stefan Steidel, steidel@mathematik.uni-kl.de

OVERVIEW:
    The algorithm of Cerlienco-Mureddu [Marinari M.G., Mora T., A remark on a
    remark by Macaulay or Enhancing Lazard Structural Theorem. Bull. of the
    Iranian Math. Soc., 29 (2003), 103-145] associates to each ordered set of
    points A:={a1,...,as} in K^n, ai:=(ai1,...,ain)@*
          - a set of monomials N and@*
          - a bijection phi: A --> N.
    Here I(A):={f in K[x(1),...,x(n)] | f(ai)=0, for all 1<=i<=s} denotes the
    vanishing ideal of A and N = Mon(x(1),...,x(n)) \ {LM(f)|f in I(A)} is the
    set of monomials which do not lie in the leading ideal of I(A) (w.r.t. the
    lexicographical ordering with x(n)>...>x(1)). N is also called the set of
    non-monomials of I(A). NOTE: #A = #N and N is a monomial basis of
    K[x(1..n)]/I(A). In particular, this allows to deduce the set of
    corner-monomials, i.e. the minimal basis M:={m1,...,mr}, m1<...<mr, of its
    associated monomial ideal M(I(A)), such that@*
    M(I(A))= {k*mi | k in Mon(x(1),...,x(n)), mi in M},@*
    and (by interpolation) the unique reduced lexicographical Groebner basis
    G := {f1,...,fr} such that LM(fi)=mi for each i, that is, I(A)=<G>.
    Moreover, a variation of this algorithm allows to deduce a canonical linear
    factorization of each element of such a Groebner basis in the sense ot the
    Axis-of-Evil Theorem by M.G. Marinari and T. Mora. More precisely, a
    combinatorial algorithm and interpolation allow to deduce polynomials
@*
@*                y_mdi = x(m) - g_mdi(x(1),...,x(m-1)),
@*
    i=1,...,r; m=1,...,n; d in a finite index-set F, satisfying
@*
@*                 fi = (product of y_mdi) modulo (f1,...,f(i-1))
@*
    where the product runs over all m=1,...,n; and all d in F.

PROCEDURES:
 nonMonomials(id);   non-monomials of the vanishing ideal id of a set of points
 cornerMonomials(N); corner-monomials of the set of non-monomials N
 facGBIdeal(id);     GB G of id and linear factors of each element of G
";

LIB "poly.lib";

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

static proc subst1(id, int m)
{
//id = poly/ideal/list, substitute the first m variables occuring in id by 1

  int i,j;
  def I = id;
  if(typeof(I) == "list")
  {
    for(j = 1; j <= size(I); j++)
    {
      for(i = 1; i <= m; i++)
      {
        I[j] = subst(I[j],var(i),1);
      }
    }
    return(I);
  }
  else
  {
    for(i = 1; i <= m; i++)
    {
      I = subst(I,var(i),1);
    }
    return(I);
  }
}

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

proc nonMonomials(id)
"USAGE:  nonMonomials(id); id = <list of vectors> or <list of lists> or <module>
         or <matrix>.@*
         Let A= {a1,...,as} be a set of points in K^n, ai:=(ai1,...,ain), then
         A can be given as@*
          - a list of vectors (the ai are vectors) or@*
          - a list of lists (the ai are lists of numbers) or@*
          - a module s.t. the ai are generators or@*
          - a matrix s.t. the ai are columns
ASSUME:  basering must have ordering rp, i.e., be of the form 0,x(1..n),rp;
         (the first entry of a point belongs to the lex-smallest variable, etc.)
RETURN:  ideal, the non-monomials of the vanishing ideal I(A) of A
PURPOSE: compute the set of non-monomials Mon(x(1),...,x(n)) \ {LM(f)|f in I(A)}
         of the vanishing ideal I(A) of the given set of points A in K^n, where
         K[x(1),...,x(n)] is equipped with the lexicographical ordering induced
         by x(1)<...<x(n) by using the algorithm of Cerlienco-Mureddu
EXAMPLE: example nonMonomials; shows an example
"
{
  list A;
  int i,j;
  if(typeof(id) == "list")
  {
    for(i = 1; i <= size(id); i++)
    {
      if(typeof(id[i]) == "list")
      {
        vector a;
        for(j = 1; j <= size(id[i]); j++)
        {
          a = a+id[i][j]*gen(j);
        }
        A[size(A)+1] = a;
        kill a;
      }
      if(typeof(id[i]) == "vector")
      {
        A[size(A)+1] = id[i];
      }
    }
  }
  else
  {
    if(typeof(id) == "module")
    {
      for(i = 1; i <= size(id); i++)
      {
        A[size(A)+1] = id[i];
      }
    }
    else
    {
      if(typeof(id) == "matrix")
      {
        for(i = 1; i <= ncols(id); i++)
        {
          A[size(A)+1] = id[i];
        }
      }
      else
      {
        ERROR("Wrong type of input!!");
      }
    }
  }

  int n = nvars(basering);
  int s;
  int m,d;
  ideal N = 1;
  ideal N1,N2;
  list Y,D,W,Z;
  poly my,t;
  for(s = 2; s <= size(A); s++)
  {

//-- compute m = max{ j | ex. i<s: pr_j(a_i) = pr_j(a_s)} ---------------------
//-- compute d = |{ a_i | i<s: pr_m(a_i) = pr_m(a_s), phi(a_i) in T[1,m+1]}| --
    m = 0;
    Y = A[1..s-1];
    N2 = N[1..s-1];
    while(size(Y) > 0)      //assume all points different (m <= size(basering))
    {
      D = Y;
      N1 = N2;
      Y = list();
      N2 = ideal();
      m++;
      for(i = 1; i <= size(D); i++)
      {
        if(A[s][m] == D[i][m])
        {
          Y[size(Y)+1] = D[i];
          N2[size(N2)+1] = N1[i];
        }
      }
    }
    m = m - 1;
    d = size(D);
    if(m < n-1)
    {
      for(i = 1; i <= size(N1); i++)
      {
        if(N1[i] >= var(m+2))
        {
          d = d - 1;
        }
      }
    }

//------- compute t = my * x(m+1)^d where my is obtained recursively --------
    if(m == 0)
    {
      my = 1;
    }
    else
    {
      W = list();
      Z = list();
      for(i = 2; i <= s-1; i++)
      {
        if((leadexp(N[i])[m+1] == d) && (coeffs(N[i],var(m+1))[nrows(coeffs(N[i],var(m+1))),1] < var(m+1)))
        {
          W[size(W)+1] = A[i];
          Z[size(Z)+1] = A[i][1..m];
        }
      }
      W[size(W)+1] = A[s];
      Z[size(Z)+1] = A[s][1..m];

      my = nonMonomials(Z)[size(Z)];
    }

    t = my * var(m+1)^d;

//---------------------------- t is added to N --------------------------------
    N[size(N)+1] = t;
  }
  return(N);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R1 = 0,x(1..3),rp;
   vector a1 = [4,0,0];
   vector a2 = [2,1,4];
   vector a3 = [2,4,0];
   vector a4 = [3,0,1];
   vector a5 = [2,1,3];
   vector a6 = [1,3,4];
   vector a7 = [2,4,3];
   vector a8 = [2,4,2];
   vector a9 = [1,0,2];
   list A = a1,a2,a3,a4,a5,a6,a7,a8,a9;
   nonMonomials(A);

   matrix MAT[9][3] = 4,0,0,2,1,4,2,4,0,3,0,1,2,1,3,1,3,4,2,4,3,2,4,2,1,0,2;
   MAT = transpose(MAT);
   print(MAT);
   nonMonomials(MAT);

   module MOD = gen(3),gen(2)-2*gen(3),2*gen(1)+2*gen(3),2*gen(2)-2*gen(3),gen(1)+3*gen(3),gen(1)+gen(2)+3*gen(3),gen(1)+gen(2)+gen(3);
   print(MOD);
   nonMonomials(MOD);

   ring R2 = 0,x(1..2),rp;
   list l1 = 0,0;
   list l2 = 0,1;
   list l3 = 2,0;
   list l4 = 0,2;
   list l5 = 1,0;
   list l6 = 1,1;
   list L = l1,l2,l3,l4,l5,l6;
   nonMonomials(L);
}

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

proc cornerMonomials(ideal N)
"USAGE:  cornerMonomials(N); N ideal
ASSUME:  N is given by monomials satisfying the condition that if a monomial is
         in N then any of its factors is in N (N is then called an order ideal)
RETURN:  ideal, the corner-monomials of the order ideal N @*
         The corner-monomials are the leading monomials of an ideal I s.t. N is
         a basis of basering/I.
NOTE:    In our applications, I is the vanishing ideal of a finte set of points.
EXAMPLE: example cornerMonomials; shows an example
"
{
  int n = nvars(basering);
  int i,j,h;
  list C;
  poly m,p;
  int Z;
  int vars;
  intvec v;
  ideal M;

//-------------------- Test: Is 1 in N ?, if no, return <1> ----------------------
  for(i = 1; i <= size(N); i++)
  {
    if(1 == N[i])
    {
      h = 1;
      break;
    }
  }
  if(h == 0)
  {
    return(ideal(1));
  }

//----------------------------- compute the set M -----------------------------
  for(i = 1; i <= n; i++)
  {
    C[size(C)+1] = list(var(i),1);
  }
  while(size(C) > 0)
  {
    m = C[1][1];
    Z = C[1][2];
    C = delete(C,1);
    vars = 0;
    v = leadexp(m);
    for(i = 1; i <= n; i++)
    {
      vars = vars + (v[i] != 0);
    }

    if(vars == Z)
    {
      h = 0;
      for(i = 1; i <= size(N); i++)
      {
        if(m == N[i])
        {
          h = 1;
          break;
        }
      }

      if(h == 0)
      {
        M[size(M)+1] = m;
      }
      else
      {
        for(i = 1; i <= n; i++)
        {
          p = m * var(i);
          for(j = 1; j <= size(C); j++)
          {
            if(p <= C[j][1] || j == size(C))
            {
              if(p == C[j][1])
              {
                C[j][2] = C[j][2] + 1;
              }
              else
              {
                if(p < C[j][1])
                {
                  C = insert(C,list(p,1),j-1);
                }
                else
                {
                  C[size(C)+1] = list(p,1);
                }
              }
              break;
            }
          }
        }
      }
    }
  }
  return(M);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x(1..3),rp;
   poly n1 = 1;
   poly n2 = x(1);
   poly n3 = x(2);
   poly n4 = x(1)^2;
   poly n5 = x(3);
   poly n6 = x(1)^3;
   poly n7 = x(2)*x(3);
   poly n8 = x(3)^2;
   poly n9 = x(1)*x(2);
   ideal N = n1,n2,n3,n4,n5,n6,n7,n8,n9;
   cornerMonomials(N);
}

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

proc facGBIdeal(id)
"USAGE:  facGBIdeal(id); id = <list of vectors> or <list of lists> or <module>
         or <matrix>.@*
         Let A= {a1,...,as} be a set of points in K^n, ai:=(ai1,...,ain), then
         A can be given as@*
          - a list of vectors (the ai are vectors) or@*
          - a list of lists (the ai are lists of numbers) or@*
          - a module s.t. the ai are generators or@*
          - a matrix s.t. the ai are columns
ASSUME:  basering must have ordering rp, i.e., be of the form 0,x(1..n),rp;
         (the first entry of a point belongs to the lex-smallest variable, etc.)
RETURN:  a list where the first entry contains the Groebner basis G of I(A)
         and the second entry contains the linear factors of each element of G
NOTE:    combinatorial algorithm due to the Axis-of-Evil Theorem of M.G.
         Marinari, T. Mora
EXAMPLE: example facGBIdeal; shows an example
"
{
  list A;
  int i,j;
  if(typeof(id) == "list")
  {
    for(i = 1; i <= size(id); i++)
    {
      if(typeof(id[i]) == "list")
      {
        vector a;
        for(j = 1; j <= size(id[i]); j++)
        {
          a = a+id[i][j]*gen(j);
        }
        A[size(A)+1] = a;
        kill a;
      }
      if(typeof(id[i]) == "vector")
      {
        A[size(A)+1] = id[i];
      }
    }
  }
  else
  {
    if(typeof(id) == "module")
    {
      for(i = 1; i <= size(id); i++)
      {
        A[size(A)+1] = id[i];
      }
    }
    else
    {
      if(typeof(id) == "matrix")
      {
        for(i = 1; i <= ncols(id); i++)
        {
          A[size(A)+1] = id[i];
        }
      }
      else
      {
        ERROR("Wrong type of input!!");
      }
    }
  }

  int n = nvars(basering);
  def S = basering;
  def R;

  ideal N = nonMonomials(A);
  ideal eN;
  ideal M = cornerMonomials(N);
  poly my, emy;

  int d,k,l,m;

  int d1;
  poly y(1);

  list N2,D,H;
  poly z,h;

  int dm;
  list Am,Z;
  ideal E;
  list V,eV;
  poly p;
  poly y(2..n),y;
  poly xi;

  poly f;
  ideal G1;          // stores the elements of G, i.e. G1 = G the GB of I(A)
  ideal Y;           // stores the linear factors of GB-elements in each loop
  list G2;           // contains the linear factors of each element of G

  for(j = 1; j <= size(M); j++)
  {
    my = M[j];
    emy = subst(my,var(1),1);         // auxiliary polynomial
    eN = subst(N,var(1),1);           // auxiliary monomial ideal
    Y = ideal();

    d1 = leadexp(my)[1];
    y(1) = 1;
    i = 0;
    k = 1;
    while(i < d1)
    {
//---------- searching for phi^{-1}(x_1^i*x_2^d_2*...*x_n^d_n) ----------------
      while(my*var(1)^i/var(1)^d1 != N[k])
      {
        k++;
      }
      y(1) = y(1)*(var(1)-A[k][1]);
      Y[size(Y)+1] = cleardenom(var(1)-A[k][1]);
      i++;
    }
    f = y(1);                             // gamma_1my

//--------------- Recursion over number of variables --------------------------
    z = 1;                                // zeta_mmy
    for(m = 2; m <= n; m++)
    {
      z = z * y(m-1);

      D = list();
      H = list();
      for(i = 1; i <= size(A); i++)
      {
        h = z;
        for(k = 1; k <= n; k++)
        {
          h = subst(h,var(k),A[i][k]);
        }
        if(h != 0)
        {
          D[size(D)+1] = A[i];
          H[size(H)+1] = i;
        }
      }

      if(size(D) == 0)
      {
        break;
      }

      dm = leadexp(my)[m];
      while(dm == 0)
      {
        m = m + 1;
        dm = leadexp(my)[m];
      }

      N2 = list();                        // N2 = N_m
      emy = subst(emy,var(m),1);
      eN = subst(eN,var(m),1);
      for(i = 1; i <= size(N); i++)
      {
        if((emy == eN[i]) && (my > N[i]))
        {
          N2[size(N2)+1] = N[i];
        }
      }

      y(m) = 1;
      xi = z;
      for(d = 1; d <= dm; d++)
      {
        Am = list();
        Z = list();                       // Z = pr_m(Am)

//------- V contains all ny*x_m^{d_m-d}*x_m+1^d_m+1*...+x_n^d_n in N_m --------
        eV = subst1(N2,m-1);
        V = list();
        for(i = 1; i <= size(eV); i++)
        {
          if(eV[i] == subst1(my,m-1)/var(m)^d)
          {
            V[size(V)+1] = eV[i];
          }
        }

//------- A_m = phi^{-1}(V) intersect D_md-1 ----------------------------------
        for(i = 1; i <= size(D); i++)
        {
          p = N[H[i]];
          p = subst1(p,m-1);
          for(l = 1; l <= size(V); l++)
          {
            if(p == V[l])
            {
              Am[size(Am)+1] = D[i];
              Z[size(Z)+1] = D[i][1..m];
              break;
            }
          }
        }

        E = nonMonomials(Z);

        R = extendring(size(E), "c(", "lp");
        setring R;
        ideal E = imap(S,E);
        list Am = imap(S,Am);
        poly g = var(size(E)+m);
        for(i = 1; i <= size(E); i++)
        {
          g = g + c(i)*E[i];
        }

        ideal I = ideal();
        poly h;
        for (i = 1; i <= size(Am); i++)
        {
          h = g;
          for(k = 1; k <= n; k++)
          {
            h = subst(h,var(size(E)+k),Am[i][k]);
          }
          I[size(I)+1] = h;
        }

        ideal sI = std(I);
        g = reduce(g,sI);

        setring S;
        y = imap(R,g);
        Y[size(Y)+1] = cleardenom(y);
        xi = xi * y;

        D = list();
        H = list();
        for(i = 1; i <= size(A); i++)
        {
          h = xi;
          for(k = 1; k <= n; k++)
          {
            h = subst(h,var(k),A[i][k]);
          }
          if(h != 0)
          {
            D[size(D)+1] = A[i];
            H[size(H)+1] = i;
          }
        }

        y(m) = y(m) * y;

        if(size(D) == 0)
        {
          break;
        }
      }

      f = f * y(m);
    }
    G1[size(G1)+1] = f;
    G2[size(G2)+1] = Y;
  }
  return(list(G1,G2));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R = 0,x(1..3),rp;
   vector a1 = [4,0,0];
   vector a2 = [2,1,4];
   vector a3 = [2,4,0];
   vector a4 = [3,0,1];
   vector a5 = [2,1,3];
   vector a6 = [1,3,4];
   vector a7 = [2,4,3];
   vector a8 = [2,4,2];
   vector a9 = [1,0,2];
   list A = a1,a2,a3,a4,a5,a6,a7,a8,a9;
   facGBIdeal(A);

   matrix MAT[9][3] = 4,0,0,2,1,4,2,4,0,3,0,1,2,1,3,1,3,4,2,4,3,2,4,2,1,0,2;
   MAT = transpose(MAT);
   print(MAT);
   facGBIdeal(MAT);

   module MOD = gen(3),gen(2)-2*gen(3),2*gen(1)+2*gen(3),2*gen(2)-2*gen(3),gen(1)+3*gen(3),gen(1)+gen(2)+3*gen(3),gen(1)+gen(2)+gen(3);
   print(MOD);
   facGBIdeal(MOD);

   list l1 = 0,0,1;
   list l2 = 0,1,-2;
   list l3 = 2,0,2;
   list l4 = 0,2,-2;
   list l5 = 1,0,3;
   list l6 = 1,1,3;
   list L = l1,l2,l3,l4,l5,l6;
   facGBIdeal(L);
}