source: git/Singular/LIB/chern.lib @ 27a5b3

////////////////////////////////////////////////////////////////
version = "version chern.lib 0.615 Feb_2015 ";      //$Id$
category = "Chern classes";
info="
LIBRARY:  chern.lib    Symbolic Computations with Chern classes

AUTHOR:  Oleksandr Iena,      oleksandr.iena@uni.lu,  yena@mathematik.uni-kl.de

REFERENCES:
  [1] Iena, Oleksandr,  On symbolic computations with Chern classes:
                        remarks on the library chern.lib for Singular,
                        http://hdl.handle.net/10993/22395, 2015.
  [2] Lascoux, Alain,   Classes de Chern d'un produit tensoriel.
                        C. R. Acad. Sci., Paris, Ser. A 286, 385-387 (1978).
  [3] Manivel, Laurent  Chern classes of tensor products, arXiv 1012.0014, 2010.

PROCEDURES:
  symm(l [,N]);             symmetric functions in the entries of l
  symNsym(f, c);            symmetric and non-symmetric parts of a polynomial f
  CompleteHomog(N, l);      complete homogeneous symmetric functions
  segre(N, c);              Segre classes in terms of Chern classes
  chern(N, s);              Chern classes in terms of Segre classes
  chNum(N, c);              the non-zero Chern numbers in degree N in the entries of c
  chNumbers(N, c);          the Chern numbers in degree N in the entries of c
  sum_of_powers(k, l);      the sum of k-th powers of the entries of l
  powSumSym(c [,N]);        the sums of powers [up to degree N] in terms
                            of the elementary symmetric polynomials (entries of l)
  chAll(c [,N]);            Chern character in terms of the Chern classes
  chAllInv(c);              Chern classes in terms of the Chern character
  chHE(c);                  the highest term of the Chern character
  ChernRootsSum(a, b);      the Chern roots of a direct sum
  chSum(c, C);              the Chern classes of a direct sum
  ChernRootsDual(l);        the Chern roots of the dual vector bundle
  chDual(c);                the Chern classes of the dual vector bundle
  ChernRootsProd(l, L);     the Chern roots of a tensor product of vector bundles
  chProd(r, c, R, C [,N]);  Chern classes of a tensor product of vector bundles
  chProdE(c, C);            Chern classes of a tensor product of vector bundles
  chProdL(r, c, R, C);      Chern classes of a tensor product of vector bundles
  chProdLP(r, c, R, C);     total Chern class of a tensor product of vector bundles
  chProdM(r, c, R, C);      Chern classes of a tensor product of vector bundles
  chProdMP(r, c, R, C);     total Chern class of a tensor product of vector bundles
  ChernRootsHom(l, L);      the Chern roots of a Hom vector bundle
  chHom(r, c, R, C [,N]);   Chern classes of the Hom-vector bundle
  ChernRootsSymm(n, l);     the Chern roots of the n-th symmetric power
                            of a vector bundle with Chern roots from l
  ChernRootsWedge(n, l);    the Chern roots of the n-th exterior power
                            of a vector bundle with Chern roots from l
  chSymm(k, r, c [,p]);     the rank and the Chern classes of the k-th symmetric power
                            of a vector bundle of rank r with Chern classes c
  chSymm2L(r, c);           the rank and the Chern classes of the second symmetric power
                            of a vector bundle of rank r with Chern classes c
  chSymm2LP(r, c);          the total Chern class of the second symmetric power
                            of a vector bundle of rank r with Chern classes c
  chWedge(k, r, c [,p]);    the rank and the Chern classes of the k-th exterior power
                            of a vector bundle of rank r with Chern classes c
  chWedge2L(r, c);          the rank and the Chern classes of the second exterior power
                            of a vector bundle of rank r with Chern classes c
  chWedge2LP(r, c);         the total Chern class of the second exterior power
                            of a vector bundle of rank r with Chern classes c
  todd(c [,n]);             the Todd class
  toddE(c);                 the highest term of the Todd class
  Bern(n);                  the second Bernoulli numbers
  tdCf(n);                  the coefficients of the Todd class of a line bundle
  tdTerms(n, f);            the terms of the Todd class of a line bundle
                            coresponding to the Chern root t
  tdFactor(n, t);           the Todd class of a line bundle coresponding
                            to the Chern root t
  cProj(n);                 the total Chern class of (the tangent bundle on)
                            the projective space P_n
  chProj(n);                the Chern character of (the tangent bundle on)
                            the projective space P_n
  tdProj(n);                the Todd class of (the tangent bundle on)
                            the projective space P_n
  eulerChProj(n, r, c);     Euler characteristic of a vector bundle on
                            the projective space P_n
                            via Hirzebruch-Riemann-Roch theorem
  chNumbersProj(n);         the Chern numbers of the projective space P_n
  classpoly(l, t);          polynomial in t with coefficients from l
                            (without constant term)
  chernPoly(l, t);          Chern polynomial (constant term 1)
  chernCharPoly(r, l, t);   polynomial in t corresponding to the Chern character
                            (constant term r)
  toddPoly(td, t);          polynomial in t corresponding to the Todd class
                            (constant term 1)
  rHRR(N, ch, td);          the main ingredient of the right-hand side
                            of the Hirzebruch-Riemann-Roch formula
  SchurS(I, S);             the Schur polynomial corresponding to partition I
                            in terms of the Segre classes S
  SchurCh(I, C);            the Schur polynomial corresponding to partition I
                            in terms of the Chern classes C
  part(m, n);               partitions of integers not exceeding n
                            into m non-negative summands
  dualPart(I [,N]);         partition dual to I
  PartC(I, m);              the complement of a partition with respect to m
  partOver(n, J);           partitions over a given partition J with summands not exceeding n
  partUnder(J);             partitions under a given partition J
";

LIB "general.lib";
LIB "lrcalc.lib"; // needed for chProdM(..) and chProdMP(..)
//----------------------------------------------------------

proc symm(list l, list #)
"USAGE:   symm(l [,n]);  l a list of polynomials, n integer
RETURN:   list of polynomials
PURPOSE:  computes the list of elementary symmetric functions in the entries of l
EXAMPLE:  example symm; shows an example
NOTE:     makes sense only for a list of polynomials
"
{
  int N=size(l);
  int n=size(l);
  if(size(#)!=0)
  {
    if( is_integer(#[1]) )
    {
      N = #[1];
    }
  }
  if(n==0) // if the list is empty, return the empty list
  {
    return(list());
  }
  else
  {
    int i, j;
    list rez=list(1, l[1]);
    for(i=2; i<=n; i++)
    {
      if( i<=N )
      {
        rez=rez+list(0);
      }
      for(j = min(i, N); j>=1; j--)
      {
        rez[j+1] = rez[j+1] + rez[j]*l[i];
      }
    }
    return(delete(rez, 1));
  }
}
example
{
  "EXAMPLE:";echo =2;
  // elementary symmetric functions in x, y, z:
  ring r = 0, (x, y, z), dp;
  list l=(x, y, z);
  print(symm(l));

  //now let us compute only the first two symmetric polynomials in a(1), ... , a(10)
  ring q= 0,(a(1..10)), dp;
  list l=a(1..10);
  print(symm(l, 2));
}
//-----------------------------------------------------------------------

proc symNsym(poly f, list c)
"USAGE:   symNsym(f, c);  f polynomial; c list of polynomials
RETURN:   list with 2 poly entries
PURPOSE:  computes a symmetric and a non-symmetric part of f
          in terms of the elementary symmetric functions from c
          as well a non-symmetric remainder
EXAMPLE:  example symNsym; shows an example
NOTE:     constants are not considered symmetric
"
{
  ideal V=variables(f); // variables f depends on
  int nV=size(V); // their number
  if(nV==0)
  {
    return(list(f, 0));
  }
  // now f is non-constant and does depend on some variables
  c=append_by_zeroes(nV, c); // append c by zeroes if it is too short
  def br@=basering; // remember the base ring
  // add additional variables to the base ring
  execute("ring r@= ("+ charstr(basering) +"),("+varstr(basering)+",c@(1..nV),A@(1..nV)), dp;" );
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  ideal V=F(V);
  poly f=F(f);
  int i;
  for(i=1; i<=nV; i++)
  {
    f=subst(f, V[i], A@(i) ); // rename the variables of f into A@(1..nV)
  }
  int N1=nvars(basering)-nV+1; // the number of variable A@(1)
  poly rez1=0; // to be the expression in c@(i) of the symmetric part of f
  poly rez2=0; // to be the remainder
  poly mon; // monomial in c@(i)
  poly monc; // the corresponding expression in A@(i)
  list l=symm(list(A@(1..nV) )); // symmetric functions in A@(i)
  intvec v=leadexp(f), 0; // the exponent of the leading monomial
  while(v[N1]!=0)
  {
    mon=leadcoef(f); // leading coefficient of f
    monc=mon;
    for(i=1; v[N1+i-1]!=0 ;i++ )
    {
      mon = mon*c@(i)^( v[N1+i-1]-v[N1+i] );
      monc = monc*l[i]^( v[N1+i-1]-v[N1+i] ); // has the same leading coefficient as f
    }
    rez1=rez1+mon; // add a monomial
    f=f-monc; // subtract the monomial
    v=leadexp(f), 0;
  }
  while( leadexp(f)!=0 )
  {
    rez2=rez2+lead(f);
    f=f-lead(f);
  }
  rez1=rez1+f;
  setring br@; // come back to the initial base ring
  // define the specialization homomorphism
  execute("map FF = r@,"+varstr(br@)+",c[1..nV], V[1..nV];");
  return( list( FF(rez1), FF(rez2) ) );
}
example
{
  "EXAMPLE:";echo=2;
  ring r=0, (x,y,z, c(1..3)), dp;
  list l=c(1..3);
  // The symmetric part of f = 3x2 + 3y2 + 3z2 + 7xyz + y
  // in terms of the elemenatary symmetric functions c(1), c(2), c(3)
  // and the remainder
  poly f = 3x2 + 3y2 + 3z2 + 7xyz + y;
  print( symNsym(f, l) );
  // Take a symmetrix polynomial in variables x and z
  f=x2+xz+z2;
  // Express it in terms of the elementary the symmetric functions
  print( symNsym(f, l)[1]);
}
//-------------------------------------------------------------------------------------------

proc CompleteHomog(int N, list c)
"USAGE:   CompleteHomog(N, c);  N integer, c list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the list of the complete homogeneous symmetric polynomials
          in terms of the elementary symmetric polynomials (entries of c)
EXAMPLE:  example CompleteHomog; shows an example
NOTE:
"
{
  c=append_by_zeroes(N, c);
  if(N<0) // if N is negative, return the empty list
  {
    return(list());
  }
  list rez=list(1); // the result will be computed here
  int i, j;
  int sign;
  poly f;
  for(i=1; i<=N; i++) // use the resursive formula
  {
    f=0;
    sign=1;
    for(j=1;j<=i; j++) // compute the next complete homogeneous symmetric polynomial
    {
      f=f+sign*c[j]*rez[i-j+1];
      sign=-sign;
    }
    rez=rez+( list(f)  );
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (x(1..3)), dp;
  list l=x(1..3);
  //Complete homogeneous symmetric polynomials up to degree 3 in variables x(1), x(2), x(3)
  print( CompleteHomog(3, l) );
}
//-----------------------------------------------------------------------

proc segre(list c, list #)
"USAGE:   segre(c[, N]); c list of polynomials, N integer
RETURN:   list of polynomials
PURPOSE:  computes the list of the Segre classes up to degree N
          in terms of the Chern classes from c
EXAMPLE:  example segre; shows an example
NOTE:
"
{
  int N;
  if(size(#)>0)
  {
    if( is_integer(#[1]) )
    {
      N=#[1];
    }
  }
  else
  {
    N=size(c);
  }
  c=append_by_zeroes(N, c);
  if(N<0) // if N is negative, return the empty list
  {
    return(list());
  }
  list rez=list(1); // the result will be computed here
  int i, j;
  poly f;
  for(i=1; i<=N; i++) // use the resursive formula
  {
    f=0;
    for(j=1;j<=i; j++) // compute the next Segre class
    {
      f=f-c[j]*rez[i-j+1];
    }
    rez=rez+( list(f)  );
  }
  return(delete(rez,1));
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (c(1..3)), dp;
  list l=c(1..3);
  //Segre classes up to degree 5 in Chern classes c(1), c(2), c(3)
  print( segre(l, 5) );
}
//-----------------------------------------------------------------------

proc chern(list s, list #)
"USAGE:   chern(s);  s list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the list of the Chern classes up to degree N
          in terms of the Segre classes from s
EXAMPLE:  example chern; shows an example
NOTE:
"
{
  return( segre(s, #) );
}
example
{
  "EXAMPLE:"; echo =2;
  ring r = 0, (s(1..3)), dp;
  list l=s(1..3);
  // Chern classes in Segre classes s(1), s(2), s(3)
  print( chern(l) );
  // This procedure is inverse to segre(...). Indeed:
  print( segre(chern(l), 3) );
}
//-----------------------------------------------------------------------

proc chNum(int N, list c)
"USAGE:   chNum(N, c);  N integer, c list
RETURN:   list
PURPOSE:  computes the Chern numbers of a vector bundle with Chern classes c
          on a complex manifold (variety) of dimension N,
          the zeroes corresponding to the higher zero Chern classes are ignored
EXAMPLE:  example chNumbers; shows an example
NOTE:     computes basically the partitions of N
          in summands not greater than the length of c
"
{
  int n=size(c);
  if(N<0)
  {
    print("");
    return( list() );
  }
  if( (n==0) || (N==0)  )
  {
    return(list(1));
  }
  if(n==1) // if there is only one entry in the list
  {
    return(list(c[1]^N));
  }
  int i;
  int j;
  poly f; // the powers of the last variable will be stored here
  list l=delete(c, n); // delete the last variable
  list L;
  list rez=chNum(N, l); // monomials not involving the last variable
  for(i=1;i<=(N div n); i++) // add the monomials involving the last variable
  {
    f=c[n]^i; // the power of the last variable
    // monomials without the last variable that,
    // multiplied by the i-th power of the last variable,
    // give a monomial of the required type
    L=chNum(N-n*i, l);
    for(j=1; j<=size(L) ;j++) // multiply every such monomial
    {
      L[j]=L[j]*f; // by the i-th power of the last variable
    }
    rez=rez+L; // add the monomials involving the i-th power of the last variable
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo=2;
  ring r = 0, (c(1..2)), dp;
  list l=c(1..2);
  // Let c(1) be a variable of degree 1, let c(2) be a variable of degree 2.
  // The monomials in c(1) and c(2) of weighted degree 5 are:
  print( chNum( 5, l ) );

  // Compare the result to the output of chNumbers(...):
  print( chNumbers(5, l) );
}
//----------------------------------------------------------------------------------------

proc chNumbers(int r, list c)
"USAGE:   chNumbers(r, c);  r integer,  c list
RETURN:   list
PURPOSE:  computes the Chern numbers of a vector bundle with Chern classes c
          on a complex manifold (variety) of dimension r
EXAMPLE:  example chNumbers; shows an example
NOTE:     computes basically the partitions of r
"
{
  if(r<0)
  {
    print("The dimension of a manifold must be a non-negative integer!");
    return(list()); // return the empty list in this case
  }
  if(r==0)
  {
    return(list(1));
  }
  //-----------------
  // from now on r>0
  //----------------
  int n=size(c);
  c=append_by_zeroes(r, c);
  c=c[1..r]; // throw away redundant data
  return(chNum(r, c));
}
example
{
  "EXAMPLE:";echo=2;
  ring r = 0, (c(1..3)), dp;
  list l=c(1..3);
  // The Chern numbers of a vector bundle with Chern classes c(1), c(2), c(3)
  // on a 3-fold:
  print( chNumbers( 3, l ) );

  // If the highest Chern class is zero, the Chern numbers are:
  l=c(1..2);
  print( chNumbers( 3, l ) );

  // Compare this to the output of chNum(...):
  print( chNum( 3, l ) );
}
//---------------------------------------------------------------------------------------

proc sum_of_powers(int k, list l)
"USAGE:   sum_of_powers(k, l);  k non-negative integer, l list of polynomials
RETURN:   polynomial
PURPOSE:  computes the sum of k-th powers of the entries of l
EXAMPLE:  example sum_of_powers; shows an example
NOTE:     returns 0 if k is negative
"
{
  if(k<0) // return 0 if k is negative
  {
    print("The exponent must be non-negative; 0 has been returned");
    return(0);
  }
  int i;
  int n=size(l);
  poly rez; // the result will be computed here
  for(i=1;i<=n;i++) // compute the sum of powers
  {
    rez=rez+l[i]^k;
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (x, y, z), dp;
  list l=x, y, z;
  //sum of 7-th powers of x, y, z
  print( sum_of_powers(7, l) );
}
//-----------------------------------------------------------------------

proc powSumSym(list c, list #)
"USAGE:   powSumSym(l [,N]);  l a list of polynomials, N integer
RETURN:   list of polynomials
PURPOSE:  computes the expressions for the sums of powers [up to degree N]
          in terms of the elementary symmetric polynomials (entries of l),
EXAMPLE:  example powSumSym; shows an example
NOTE:     returns the terms of the Chern character
          multiplied by the correspoding factorials
"
{
  int n;
  if( size(#) == 0 ) // if there are no optional parameters
  {
    n = size(c); // set n to be the length of c
  }
  else // if there are optional parameters
  {
    if( is_integer(#[1])) // if the first optional parameter is an integer
    {
      n = max( #[1], 0 ); // if the parameter is negative, reset it to be zero
      c = append_by_zeroes(n, c); // if n is greater than the length of c, append c by zeroes
      if( n != 0 ) // if n is non-zero
      {
        c = c[1..n]; // take into account only the first n entries of c
      }
    }
    else // if the optional parameter is not an integer, then
    {
      n=size(c); // ingore it and set n to be the length of c
    }
  }
  list rez; // the result will be computed here
  if(n==0) // return the empty list
  {
    return(rez)
  }
  else // otherwise proceed as follows:
  {
    // first compute the sums of powers of the Chern roots
    // in terms of the Chern classes using the Newton's identities
    int i, j, sign;
    poly f;
    // the first term of the Chern character coincides with the first Chern class,
    // or equivalently with the sum of Chern roots
    rez = rez + list(c[1]);
    // compute the sums of powers of Chern roots recursively using the Newton's identities
    for(j=2; j<=n; j++)
    {
      sign=1;
      f=0;
      for(i=1; i<j; i++)
      {
        f=f + c[i]*sign*rez[j-i];
        sign = -sign;
      }
      f=f+sign*j*c[j];
      rez=rez+list(f); // add the newly computed sum of powers of Chern roots to the list
    }
    return(rez); // return the result
  }
}
example
{
  "EXAMPLE:";echo =2;
  // the expressions of the first 3 sums of powers of 3 variables a(1), a(2), a(3)
  // in terms of the elementary symmetric polynomials c(1), c(2), c(3):
  ring r = 0, (c(1..3)), dp;
  list l=(c(1..3));
  print(powSumSym(l));
  // The first 5 sums in the same situation
  print(powSumSym(l, 5));
}
//---------------------------------------------------------------------------------

proc chAll(list c, list #)
"USAGE:   chAll(l [,N]);  l a list of polynomials, N integer
RETURN:   list of polynomials
PURPOSE:  computes the list of terms of positive degree [up to degree N] of
          the Chern character, where the entries of l are considered as the Chern classes
EXAMPLE:  example chAll; shows an example
NOTE:     makes sense only for a list of polynomials
"
{
    list rez; // to be the result
    rez = powSumSym(c, #); // get the sums of powers of the Chern roots
    int n = size(rez);
    bigint fct=1;
    int i;
    for(i=1;i<=n;i++) // get the terms of the Chern character
    {
      fct=fct*i;
      rez[i]=rez[i]/fct;
    }
    return(rez); // return the result
}
example
{
  "EXAMPLE:";echo =2;
  // Chern character (terms of degree 1, 2, 3)
  // corresponding to the Chern classes c(1), c(2), c(3):
  ring r = 0, (c(1..3)), dp;
  list l=(c(1..3));
  print(chAll(l));
  // terms up to degree 5 in the same situation
  print(chAll(l, 5));
}
//---------------------------------------------------------------------------------

proc chAllInv(list c)
"USAGE:   chAllInv(l);  l a list of polynomials
RETURN:   list of polynomials
PURPOSE:  procedure inverse to chAll(), computes the list of Chern classes
          from the list of terms of positive degree of the Chern character
EXAMPLE:  example chAllInv; shows an example
NOTE:     makes sense only for a list of polynomials
"
{
  int n = size(c);
  list rez;
  if(n==0) // if the list of terms of Chern character is empty, return the empty list
  {
    return(rez);
  }
  else // otherwise compute the result using the Newton's identities
  {
    int j, i, sign;
    poly f;
    // transform the list of terms of the Chern character
    // to the list of sums of powers of Chern roots
    //bigint fct=1;
    for(i=1; i<=n; i++)
    {
      //fct=fct*i;
      //c[i]=fct*c[i];
      c[i]=factorial(i)*c[i];
    }
    // the first Chern class coincides with the first degree term of the Chern character
    rez=rez+list(c[1]);
    // compute the higher Chern classes recursively using the Newton's identities
    for(j=2;j<=n;j++)
    {
      sign=1;f=0;
      for(i=1;i<j;i++)
      {
        f=f+ c[i]*sign*rez[j-i];
        sign=-sign;
      }
      f=f+sign*c[j];
      rez=rez+list(f/j);
    }
    return(rez); // return the result
  }
}
example
{
  "EXAMPLE:";echo=2;
  // first 3 Chern classes in terms of the first 3 terms
  // of the Chern character Chern  ch(1), ch(2), ch(3):
  ring r = 0, (ch(1..3)), dp;
  list l=(ch(1..3));
  print(chAllInv(l));
  // let's see that chAllInv() is inverse to chAll()
  print( chAll( chAllInv(l) ) );
}
//---------------------------------------------------------------------------------

proc chHE(list c)
"USAGE:   chHE(c);  c list of polynomials
RETURN:   polynomial
PURPOSE:  computes the highest relevant term of the Chern character
EXAMPLE:  example chHE; shows an example
NOTE:     uses the elimination and is extremely inefficient,
          is included just for comparison with chAll(c)
"
{
  int i;
  // insure that the entries of c are polynomials
  // in order to be able to apply maps
  for(i=1;i<=size(c);i++)
  {
    c[i]=poly(c[i]);
  }
  int n=size(c);
  if(n==0) // in this case return the empty list
  {
    return(list());
  }
  else // otherwise proceed as follows
  {
    def br@=basering; // remember the base ring
    // add additional variables c@, a@(1..n) to the base ring
    execute("ring r@= (" + charstr(basering) + "),(c@,"+varstr(basering)+", a@(1..n)), dp;" );
    execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
    list c=F(c); // embedd c in the bigger ring
    poly rez;
    list A=a@(1..n);
    list sym=symm(A);
    ideal I;
    poly E=1; // to be the product of variables which should be eliminated
    for(i=1;i<=n;i++)
    {
      E=E*a@(i); // compute the product of the variables that must be eliminated
      I=I, c[i]-sym[i];
    }
    I=I, c@-sum_of_powers(n, A);
    I=elim(I, E);
    rez = -subst(I[1], c@, 0);
    setring br@; // come back to the initial base ring
    execute( "map FF= r@,0,"+varstr(br@)+";" ); // define the specialization homomorphism
    poly rez=FF(rez); // bring the result to the base ring
    return( (1/factorial(n))*rez);
  }
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (c(1..3)), dp;
  list l=c(1..3);
  //the third degree term of the Chern character
  print( chHE(l) );
}
//----------------------------------------------------------------------

proc ChernRootsSum(list a, list b)
"USAGE:   ChernRootsSum(a, b); a, b lists of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern roots of the direct (Whitney) sum
          of a vector bundle with Chern roots a and a vector bundle with Chern roots b
EXAMPLE:  example ChernRootsSum; shows an example
NOTE:
"
{
  return(a+b);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (a(1..3), b(1..2)), dp;
  // assume a(1), a(2), a(3) are the Chern roots of a vector bundle E
  // assume b(1), b(2) are the Chern roots of a vector bundle F
  list l=a(1..3);
  list L=b(1..2);
  // the Chern roots of their direct sum is
  print( ChernRootsSum(l, L) );
}
//----------------------------------------------------------------------

proc chSum(list c, list C)
"USAGE:   chSum(c, C);  c, C lists of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern classes of a direct sum of two vector bundles
EXAMPLE:  example chSum; shows an example
NOTE:
"
{
  int N=size(c)+size(C);
  c=append_by_zeroes(N, c); // append by zeroes if necessary
  C=append_by_zeroes(N, C); // append by zeroes if necessary
  list rez; // to be the result
  int i;
  int j;
  poly f;
  for(i=1;i<=N;i++)
  {
    f=c[i]+C[i];
    for(j=1;j<i;j++)
    {
      f=f+c[j]*C[i-j];
    }
    rez=rez+list(f);
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (c(1..3), C(1..2)), dp;
  // Let E be a vector bundle with Chern classes c(1), c(2), c(3).
  // Let F be a vector bundle with Chern classes C(1), C(2).
  list l=c(1..3);
  list L=C(1..2);
  // Then the Chern classes of their direct sum are
  print( chSum(l, L) );
}
//----------------------------------------------------------------------

proc ChernRootsDual(list l)
"USAGE:   ChernRootsDual(l); l a list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern roots of the dual vector bundle
          of a vector bundle with Chern roots from l
EXAMPLE:  example ChernRootsDual; shows an example
NOTE:
"
{
  int n=size(l);
  int i;
  for(i=1;i<=n;i++) // change the sign of the entries of a
  {
    l[i]=-l[i];
  }
  return(l);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (a(1..3)), dp;
  // assume a(1), a(2), a(3) are the Chern roots of a vector bundle
  list l=a(1..3);
  // the Chern roots of the dual vector bundle
  print( ChernRootsDual(l) );
}
//----------------------------------------------------------------------

proc chDual(list c)
"USAGE:   chDual(c);   c list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the list of Chern classes of the dual vector bundle
EXAMPLE:  example chDual; shows an example
NOTE:
"
{
  int n=size(c);
  int i;
  for(i=1;i<=n;i=i+2)
  {
    c[i]=-c[i];
  }
  return(c);
}
example
{
  "EXAMPLE:"; echo=2;
  // Chern classes of a vector bundle that is dual to a vector bundle
  // with Chern classes c(1), c(2), c(3)
  ring r=0, (c(1..3)), dp;
  list l=c(1..3);
  print(chDual(l));
}
//-----------------------------------------------------------------------------------------

proc ChernRootsProd(list a, list b)
"USAGE:   ChernRootsProd(a, b); a, b lists of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern roots of the tensor product of a vector bundle with Chern roots a
          and a vector bundles with Chern roots b
EXAMPLE:  example ChernRootsProd; shows an example
NOTE:
"
{
  int na=size(a);
  int nb=size(b);
  int i;
  int j;
  list rez; // the result will be computed here
  for(i=1;i<=na;i++) // compute the result
  {
    for(j=1;j<=nb;j++)
    {
      rez=rez+list(a[i]+b[j]);
    }
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo=2;
  ring r=0, (a(1..2), b(1..3)), dp;
  list l=a(1..2);
  list L=b(1..3);
  // Chern roots of the tensor product of a vector bundle with Chern roots a(1), a(2)
  // and a vector bundle with Chern roots b(1), b(2), b(3)
  print(ChernRootsProd(l, L));
}
//-----------------------------------------------------------------------------------------

proc chProd(def r,  list c, def R, list C, list #)
"USAGE:   chProd(r, c, R, C [, N]);  r, R polynomials (integers);
          c, C lists of polynomials, N integer
RETURN:   list of polynomials
PURPOSE:  computes the list of Chern classes of the product of two vector bundles
          in terms of their ranks and Chern clases [up to degree N]
EXAMPLE:  example chProd; shows an example
NOTE:
"
{
  // check the input data
  if( is_integer(r) ) // if r is an integer
  {
    if(r<=0) // if r is negative or zero return the empty list
    {
      return( list() );
    }
    //----------------------------
    //now r is a positive integer
    //----------------------------
    c=append_by_zeroes(r, c); // append c by zeroes if r is greater than the length of c
    c=c[1..r]; // make c shorter (of length r) if r is smaller than the length of c
  }
  if( is_integer(R) ) // if R is an integer
  {
    if(R<=0) // if R is negative or zero return the empty list
    {
      return( list() );
    }
    //----------------------------
    //now R is a positive integer
    //----------------------------
    C=append_by_zeroes(R, C); // append C by zeroes if R is greater than the length of C
    C=C[1..R]; // make C shorter (of length R) if R is smaller than the length of C
  }
  //----------------------------------------------------------
  //now r > 0 if it is an integer; R > 0 if it is an integer
  //----------------------------------------------------------
  int n;
  if( is_integer(r) && is_integer(R) ) // if both r and R are integers
  {
    n=r*R; // set n to be the rank of the product bundle
  }
  else // otherwise define the rank of the product vector bundle by
  {
    n=size(c)*size(C); // looking at the lenghts of c and C
  }
  if( size(#) != 0 ) // if there is an optional parameter
  {
    if( is_integer( #[1] )  ) // if this parameter is an integer
    {
      if( #[1]<=0 ) // if it is negative or zero, return the empty list
      {
        return( list() );
      }
      // now #[1] is positive
      // the product bundle can only have non-zero Chern classes up to degree n
      // so ignore the optional parameter if it is greater than n
      n = min(#[1], n);
    }
  }
  if(n==0) // if n is zero, return the empty list
  {
    return( list() );
  }
  //-----------------------------------------------------------
  //now n is positive, we can perform the relevant computations
  //-----------------------------------------------------------
  int i, j;
  c=append_by_zeroes(n, c); // append c by zeroes up to degree n
  C=append_by_zeroes(n, C); // append C by zeroes up to degree n
  c=c[1..n]; // throw away the redundant data if needed
  C=C[1..n]; // throw away the redundant data if needed
  // build the list of all terms of the Chern characters: for rank r, and Chern classes c
  list ch = list(r) + chAll(c);
  list CH = list(R) + chAll(C); // do the same for rank R and Chern classes C
  poly f;
  list chP;
  // compute the list of the non-zero degree terms of the Chern character
  // of the tensor product of two vector bundles
  for(i=1;i<=n;i++) // using the multiplicativity of the Chern character
  {
    f=0;
    for(j=0;j<=i;j++)
    {
      f=f+ch[j+1]*CH[i-j+1];
    }
    chP=chP+list(f);
  }
  return( chAllInv(chP) ); // return the corresponding Chern classes
}
example
{
  "EXAMPLE:"; echo =2;
  ring H = 0, ( r, R, c(1..3), C(1..2) ), dp;
  list l=c(1..3);
  list L=C(1..2);
  // the Chern classes of the tensor product of a vector bundle E of rank 3
  // with Chern classes c(1), c(2), c(3)
  // and a vector bundle F of rank 2 with Chern classes C(1) and C(2):
  print( chProd(3, l, 2, L) );
  // the first two Chern classes of the tensor product
  // of a vector bundle E of rank r with Chern classes c(1) and c(2)
  // and a vector bundle G of rank R with Chern classes C(1) and C(2)
  // this gives the Chern classes of a tensor product on a complex surface
  l=c(1..2);
  L=C(1..2);
  print( chProd(r, l, R, L, 2 ) );
}
//---------------------------------------------------------------------------------

proc chProdE(list c, list C)
"USAGE:   chProdE(c, C);   c, C lists of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the list of Chern classes of the product
          of two vector bundles in terms of their Chern clases
EXAMPLE:  example chProdE; shows an example
NOTE:     makes sense only for (lists of) polynomials;
          uses elimination, hence very inefficient;
          included only for comparison with chProd(...)
"
{
  int r=size(c);
  int R=size(C);
  // insure that the entries of c and C are polynomials
  // in order to be able to apply maps
  int i,j;
  for(i=1;i<=r;i++)
  {
    c[i]=poly(c[i]);
  }
  for(i=1;i<=R;i++)
  {
    C[i]=poly(C[i]);
  }
  if( (r==0) && (R==0) ) // if one of the ranks is 0,
  {
    return( list() ); // return the empty list (zero bundles have no Chern classes)
  }
  //------------------------------------
  //now both r and R are greater than 0
  //------------------------------------
  int n=r*R; // the rank of the product of two vector bundles
  def br@=basering; // remember the base ring
  // add additional variables a@(1..r), b@(1..R), x@ to the base ring
  execute("ring r@=("+ charstr(basering) +"),(x@,"+varstr(basering)+",a@(1..r),b@(1..R)), dp;");
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c=F(c); // embedd c in the bigger ring
  list C=F(C); // embedd C in the bigger ring
  list A=a@(1..r); // list of Chern roots of the first vector bundle
  list syma = symm(A); // symmetric functions in the Chern roots of the first vector bundles
  list B=b@(1..R); // list of Chern roots of the second vector bundle
  list symb=symm(B); // symmetric functions in the Chern roots of the second vector bundles
  ideal I;
  // the product of variables (all Chern roots) which should be eliminated
  poly E=product(A)*product(B);
  for(i=1; i<=r; i++)
  {
    for(j=1; j<=R; j++)
    {
      I=I, c[i]-syma[i], C[j]-symb[j]; // add the relations
    }
  }
  // the Chern roots of the tensor product in terms of the Chern roots of the factors
  list crt=ChernRootsProd(A, B);
  list Cf=symm(crt); // Chern classes of the product in terms of the Chern roots of the factors
  list rez; // the result will be computed here
  ideal J;
  for(i=1;i<=n;i++)
  {
    J = I, x@-Cf[i]; // add the equation for the i-th Chern class to the ideal of relations
    J = elim(J, E); // eliminate the Chern roots
    // get the expression for the i-th Chern class of the product
    // in terms of the Chern classes of the factors
    rez = rez + list( -subst(J[1], x@, 0) );
  }
  setring br@; // come back to the initial base ring
  execute( "map FF= r@, 0,"+varstr(br@)+";" ); // define the specialization homomorphism t@=0
  list rez=FF(rez); // bring the result to the base ring
  return(rez); // return the corresponding Chern classes
}
example
{
  "EXAMPLE:"; echo =2;
  ring H = 0, ( c(1..3), C(1..2) ), dp;
  list l=c(1..3);
  list L=C(1..2);
  // the Chern classes of the tensor product of a vector bundle E of rank 3
  // with Chern classes c(1), c(2), c(3)
  // and a vector bundle F of rank 2 with Chern classes C(1) and C(2):
  print( chProdE(l,  L) );
}
//------------------------------------------------------------------------------------

proc chProdL(int r, list c, int R, list C)
"USAGE:   chProdL(r, c, R, C);  r, R integers; c, C lists of polynomials
RETURN:   list
PURPOSE:  computes the list of Chern classes of the product of two vector bundles
          in terms of their Chern clases
EXAMPLE:  example chProdL; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  // check the input data
  if(r<=0) // if r is negative or zero return the empty list
  {
    return( list() );
  }
  //----------------------------
  //now r is a positive integer
  //----------------------------
  c=append_by_zeroes(r, c); // append c by zeroes if r is greater than the length of c
  c=c[1..r]; // make c shorter (of length r) if r is smaller than the length of c
  if(R<=0) // if R is negative or zero return the empty list
  {
    return( list() );
  }
  //----------------------------
  //now R is a positive integer
  //----------------------------
  C=append_by_zeroes(R, C); // append C by zeroes if R is greater than the length of C
  C=C[1..R]; // make C shorter (of length R) if R is smaller than the length of C
  //----------------------------------------------------------
  // now r > 0 and R > 0
  //----------------------------------------------------------
  def br@=basering; // remember the base ring
  // add  additional variables to the base ring
  execute("ring r@=("+charstr(basering)+"), ("+varstr(basering)+", t@, c@(1..r), C@(1..R)), dp;");
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c, C;
  int i;
  for(i=1;i<=r;i++)
  {
    c[i]=c@(i)*t@^i;
  }
  for(i=1;i<=R;i++)
  {
    C[i]=C@(i)*t@^i;
  }
  poly f = chProdLP(r,c,R,C); // get the total Chern class using the Lascoux formula
  matrix CF = coeffs(f, t@); // get its coefficients in front of the powers of t@
  int N=r*R;
  list rez; // write them in a list
  for(i=1;i<=N;i++)
  {
    rez=rez+list(CF[i+1,1]);
  }
  setring br@; // come back to the initial base ring
  // define the specialization homomorphism
  execute("map FF = r@,"+varstr(br@)+",0, c[1..r], C[1..R];");
  return( FF( rez ) ); // bring the result to the initial ring
}
example
{
  "EXAMPLE:"; echo =2;
  // The Chern classes of the tensor product of a vector bundle of rank 3
  // with Chern classes c(1), c(2), c(3) and a vector bundle of rank 1 with
  // Chern class C(1)
  ring r = 0, ( c(1..3), C(1)), dp;
  list c=c(1..3);
  list C=C(1);
  print( chProdL(3,c,1,C) );
}
//---------------------------------------------------------------------------------------

proc chProdLP(int r, list c, int R, list C)
"USAGE:   chProdLP(r, c, R, C);  r, R integers; c, C lists of polynomials
RETURN:   polynomial
PURPOSE:  computes the total Chern class of the product of two vector bundles
          in terms of their ranks and Chern clases
EXAMPLE:  example chProdLP; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  if(r<=0) // if r is negative or zero, return 1
  {
    return( 1 );
  }
  if(R<=0) // if R is negative or zero, return 1
  {
    return( 1 );
  }
  //-------------------------------------------
  // now r and R are positive
  //-------------------------------------------
  c=append_by_zeroes(r, c);
  C=append_by_zeroes(R, C);
  c=c[1..r];
  C=C[1..R];
  list P;
  P=part(r, R); // compute the partitions of numbers up to R into r summands
  int sz=size(P); // number of such partitions
  int szu;
  int i, j;
  list T;
  list PU;
  list TU;
  poly rez; // the result will be computed here
  poly ST;
  // implement the formula of Lascoux:
  for(i=1;i<=sz;i++) // run through all the partitions from P
  {
    T=P[i]; // the current partition
    ST= SchurS( PartC(T, R) , C ); // compute the corresponding Schur polynomial
    PU=partUnder(T); // compute the partitions under T
    szu=size(PU); // number of such partitions
    for(j=1;j<=szu;j++) // run through all the partitions lying under T
    {
      TU=PU[j]; // for each of them
      rez=rez+IJcoef(T, TU)* SchurCh(TU, c) *ST; // add the corresponding term to the result
     }
  }
  return(rez); // return the result
}
example
{
  "EXAMPLE:"; echo =2;
  // The total Chern class of the tensor product of a vector bundle of rank 3
  // with Chern classes c(1), c(2), c(3) and a vector bundle of rank 1 with
  // Chern class C(1)
  ring r = 0, ( c(1..3), C(1)), ws(1,2,3, 1);
  list c=c(1..3);
  list C=C(1);
  print( chProdLP(3,c,1,C) );
}
//---------------------------------------------------------------------------------------

proc chProdM(int r, list c, int R, list C)
"USAGE:   chProdM(r, c, R, C);  r, R integers; c, C lists of polynomials
RETURN:   list
PURPOSE:  computes the list of Chern classes of the product of two vector bundles
          in terms of their Chern clases
EXAMPLE:  example chProdM; shows an example
NOTE:     Implementation of the formula of Manivel
"
{
  // check the input data
  if(r<=0) // if r is negative or zero return the empty list
  {
    return( list() );
  }
  //----------------------------
  //now r is a positive integer
  //----------------------------
  c=append_by_zeroes(r, c); // append c by zeroes if r is greater than the length of c
  c=c[1..r]; // make c shorter (of length r) if r is smaller than the length of c
  if(R<=0) // if R is negative or zero return the empty list
  {
    return( list() );
  }
  //----------------------------
  //now R is a positive integer
  //----------------------------
  C=append_by_zeroes(R, C); // append C by zeroes if R is greater than the length of C
  C=C[1..R]; // make C shorter (of length R) if R is smaller than the length of C
  //----------------------------------------------------------
  // now r > 0 and R > 0
  //----------------------------------------------------------
  def br@=basering; // remember the base ring
  // add  additional variables to the base ring
  execute("ring r@=("+charstr(basering)+"), ("+varstr(basering)+", t@, c@(1..r), C@(1..R)), dp;");
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c, C;
  int i;
  for(i=1;i<=r;i++)
  {
    c[i]=c@(i)*t@^i;
  }
  for(i=1;i<=R;i++)
  {
    C[i]=C@(i)*t@^i;
  }
  poly f = chProdMP(r,c,R,C); // get the total Chern class using the Manivel formula
  matrix CF = coeffs(f, t@); // get its coefficients in front of the powers of t@
  int N=r*R;
  list rez; // write them in a list
  for(i=1;i<=N;i++)
  {
    rez=rez+list(CF[i+1,1]);
  }
  setring br@; // come back to the initial base ring
  // define the specialization homomorphism
  execute("map FF = r@,"+varstr(br@)+",0, c[1..r], C[1..R];");
  return( FF( rez ) ); // bring the result to the initial ring
}
example
{
  "EXAMPLE:"; echo = 2;
  // The Chern classes of the tensor product of a vector bundle of rank 3
  // with Chern classes c(1), c(2), c(3) and a vector bundle of rank 1 with
  // Chern class C(1)
  ring r = 0, ( c(1..3), C(1)), dp;
  list c=c(1..3);
  list C=C(1);
  print( chProdM(3,c,1,C) );
}
//---------------------------------------------------------------------------------------

proc chProdMP(int r, list c, int R, list C)
"USAGE:   chProdMP(r, c, R, C);  r, R integers; c, C lists of polynomials
RETURN:   polynomial
PURPOSE:  computes the total Chern class of the product of two vector bundles
          in terms of their ranks and Chern clases
EXAMPLE:  example chProdMP; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  if(r<=0) // if r is negative or zero, return 1
  {
    return( 1 );
  }
  if(R<=0) // if R is negative or zero, return 1
  {
    return( 1 );
  }
  //-------------------------------------------
  // now r and R are positive
  //-------------------------------------------
  c=append_by_zeroes(r, c);
  C=append_by_zeroes(R, C);
  c=c[1..r];
  C=C[1..R];
  list P;
  P=part(r, R); // compute the partitions of numbers up to R into r summands
  int sz=size(P); // number of such partitions
  int szu;
  int i, j;
  list T;
  list PU;
  list TU;
  poly rez; // the result will be computed here
  poly ST;
  // implement the formula of Manivel:
  for(i=1;i<=sz;i++) // run through all the partitions from P
  {
    T=P[i]; // the current partition
    ST= SchurS( PartC(T, R) , C ); // compute the corresponding Schur polynomial
    PU=partUnder(T); // compute the partitions under T
    szu=size(PU); // number of such partitions
    for(j=1;j<=szu;j++) // run through all the partitions lying under T
    {
      TU=PU[j]; // for each of them
      // add the corresponding term to the result
      rez=rez+Pcoef( TU, PartC(dualPart(T, R), r), r, R )* SchurCh(TU, c) *ST;
    }
  }
  return(rez); // return the result
}
example
{
  "EXAMPLE:"; echo =2;
  // The total Chern class of the tensor product of a vector bundle of rank 3
  // with Chern classes c(1), c(2), c(3) and a vector bundle of rank 1 with
  // Chern class C(1)
  ring r = 0, ( c(1..3), C(1)), ws(1,2,3, 1);
  list c=c(1..3);
  list C=C(1);
  print( chProdMP(3,c,1,C) );
}
//---------------------------------------------------------------------------------------

proc ChernRootsHom(list a, list b)
"USAGE:   ChernRootsHom(a, b); a, b lists of polynomials
RETURN:   list of polynomials
PURPOSE:  for a vector bundle E with Chern roots a and a vector bundle F
          with Chern roots b, computes the Chern roots of Hom(E, F)
EXAMPLE:  example ChernRootsHom; shows an example
NOTE:
"
{
  int na=size(a);
  int nb=size(b);
  int i;
  int j;
  list rez; // the result will be computed here
  for(i=1;i<=na;i++) // compute the result
  {
    for(j=1;j<=nb;j++)
    {
      rez=rez+list(-a[i]+b[j]);
    }
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo=2;
  ring r=0, (a(1..2), b(1..3)), dp;
  list l=a(1..2);
  list L=b(1..3);
  // Let E be a vector bundle with Chern roots a(1). a(2),
  // let F be a vector bundle with CHern roots b(1), b(2), b(3).
  // Then the Chern roots of Hom(E, F) are
  print(ChernRootsHom(l, L));
}
//-----------------------------------------------------------------------------------------

proc chHom(def r,  list c, def R, list C, list #)
"USAGE:   chHom(r, c, R, C [, N]);  r, R polynomials (integers);
          c, C lists of polynomials, N integer
RETURN:   list of polynomials
PURPOSE:  computes [up to degree N] the list of Chern classe of the vector bundle Hom(E, F)
          in terms of the ranks and the Chern clases of E and F
EXAMPLE:  example chHom; shows an example
NOTE:
"
{
  return( chProd(r, chDual(c), R, C, # ) );
}
example
{
  "EXAMPLE:"; echo=2;
  ring H = 0, ( r, R, c(1..3), C(1..2) ), dp;
  list l=c(1..3);
  list L=C(1..2);
  // the Chern classes of Hom(E, F) for a vector bundle E of rank 3
  // with Chern classes c(1), c(2), c(3)
  // and a vector bundle F of rank 2 with Chern classes C(1) and C(2):
  print( chHom(3, l, 2, L) );
  // the first two Chern classes of Hom(E, F) for a vector bundle E of rank r
  // with Chern classes c(1) and c(2)
  // and a vector bundle G of rank R with Chern classes C(1) and C(2)
  // this gives the Chern classes of a tensor product on a complex surface
  l=c(1..2);
  L=C(1..2);
  print( chHom(r, l, R, L, 2 ) );
}
//---------------------------------------------------------------------------------

proc ChernRootsSymm(int n, list l)
"USAGE:   ChernRootsSymm(m, l); m integer, l a list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern roots of m-th symmetric power
          of a vector bundle with Chern roots from l
EXAMPLE:  example ChernRootsSymm; shows an example
NOTE:
"
{
  if(n<0) // return the empty list if n is negative
  {
    return(list(0));
  }
  int r=size(l);
  def br@=basering; // remember the base ring
  ring r@=0, (a@(1..r)), dp;
  ideal mon = a@(1..r);
  mon=mon^n; // all monomials of degree n
  list rez;
  int i, j;
  int N = size(mon);
  intvec v;
  for(i=1; i<=N; i++) // collect in rez the exponents of the monomials of degree n
  {
    v = leadexp(mon[i]);
    rez = rez + list(v);
  }
  setring br@;
  poly f;
  list rez1;
  // run over all exponents and construct the corresponding sums of the Chern roots
  for(i=1; i<=N; i++)
  {
    f=0;
    for(j=1;j<=r;j++)
    {
      f=f+rez[i][j]*l[j];
    }
    rez1=rez1+list(f);
  }
  return(rez1);
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (a(1..3)), dp;
  list l=a(1..3);
  // the Chern roots of the second symmetric power of a vector bundle
  // with Chern  roots a(1), a(2), a(3)
  print( ChernRootsSymm(2, l) );
}
//------------------------------------------------------------

proc ChernRootsWedge( int m, list l)
"USAGE:   ChernRootsWedge(m, l); m integer, l a list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern roots of m-th exterior power
          of a vector bundle with Chern roots from l
EXAMPLE:  example ChernRootsWedge; shows an example
NOTE:     makes sense only for list of polynomials
"
{
  int n=size(l);
  if((m>n)||(m<=0) ) // if m is bigger that n or non-positive
  {
    return( list(0) ); // return the list with one zero entry
  }
  else
  {
    if(m==n) // if m equals n, the only Chern root of the exterior power will be
    {
      return( list(sum(l)) ); // the sum of the initial Chern roots
    }
    else // otherwise proceed recursively
    {
      int i;
      list rez;
      list rez1;
      list l1 = delete(l, 1); // throw away the first element from the list
      poly f = l[1]; // remember the first entry of l
      // compute the Chern roots of the (m-1)-th exterior power of the smaller list
      rez1 = ChernRootsWedge(m-1, l1 );
      int s = size( rez1 );
      // add the first entry of the bigger list to every entry in the result,
      // this will give all Chern roots involving f
      for(i=1; i<=s; i++)
      {
        rez1[i] = f+rez1[i];
      }
      // return the union of those Chern roots with f and those without f
      rez = ChernRootsWedge(m, l1) + rez1;
      return( rez );
    }
  }
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (a(1..3)), dp;
  list l=a(1..3);
  // the Chern roots of the second exterior power of a vector bundle
  // with Chern  roots a(1), a(2), a(3)
  print( ChernRootsWedge(2, l) );
}
//---------------------------------------------------------------------------------

proc chSymm(int k, int r, list c, list #)
"USAGE:   chSymm(k, r, c[, pos]);  k, r integers, c list of polynomials, pos list of integers
RETURN:   list with entries: int N, list of polynomials l
PURPOSE:  computes the rank and the Chern classes of the symmetric power of a vector bundle
EXAMPLE:  example chSymm; shows an example
NOTE:     for the second symmetric power chSymm2L(...) could be faster
"
{
  // insure that the entries of c are polynomials
  // in order to be able to apply maps
  int i;
  for(i=1;i<=size(c);i++)
  {
    c[i]=poly(c[i]);
  }
  if(r<0) // if the rank is negative
  {
    print("The rank of a vector bundle can non be negative");
    return(list()); // return the empty list in this case
  }
  if(r==0) // if we deal with the zero bundle
  {
    return( list( 0, list() ) ); // return the data corresponding to the zero bundle
  }
  //-----------------------------------
  // from now on we are in the case r>0
  //-----------------------------------
  // if the length  n of the list of Chern classes is smaller
  // than the rank of the vector bundle,
  // the higher classes are assumed to be zero and the list is appended by zeroes up to length r
  c=append_by_zeroes(r, c);
  // if the lenght of the list of the Chern classes is greater than the rank
    c=c[1..r]; // throw away the redundant data
  //-----------------------------------
  // from now on the lenght of c is r>0
  //-----------------------------------
  if(k<0)
  {
    print("You are trying to compute a negative symmetric power of a vector bundle");
    return( list(0, list() ) ); // assume such a power to be just a zero bundle
  }
  if(k==0) // the zeroth symmetric power is the trivial line bundle
  {
    return( list(1, list(0)) );
  }
  if(k==1) // the first symmetric power is equal to the vector bundle itself
  {
    return(list(r, c));
  }
  //-----------------------------------
  // from now on we are in the case k>2
  //-----------------------------------
  list LM = integer_list(#);
  int M = LM[2]; // maximum among the optional parameters
  # = LM[1]; // take into account only the first integer optional parameters that are positive
  //-------------------------------
  // Perform the computations now
  //-------------------------------
  def br@=basering; // remember the base ring
  // add additional variables to the base ring
  execute("ring r@=(" + charstr(basering) +  "),(x@,"+varstr(basering)+", a@(1..r)), dp;" );
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c=F(c); // embed c into the bigger ring
  list rez; // the Chern classes of the symmetric power are going to be written here
  poly E = product( list( a@(1..r ) )  ); // product of the Chern roots
  list ss=ChernRootsSymm(k, list( a@(1..r) ) ); // list of the Chern roots of the symmetric power
  int N=size(ss); // number of such roots, it equals the rank of the symmetric power
  // the entries in C will be the Chern classes of the symmetric power
  // expressed in terms of the Chern roots of the initial vector bundle
  list C;
  ideal I, J;
  // list of the Chern classes of the initial vector bundle expressed in its Chern roots
  list sym=symm(list(a@(1..r)));
  if(size(#)==0) // if there are no optional parameters, compute all Chern classes
  {
    // the entries here are the Chern classes of the symmetric power
    // expressed in terms of Chern roots of the initial vector bundle
    C=symm(ss);
    for(i=1;i<=N;i++) // eliminate the Chern roots
    {
      if(i<= r) // first add all relevant  formulas for the Chern classes in terms of Chern roots
      {
        I=I, c[i]-sym[i];
      }
      J = I,  x@-C[i];
      // Notice that elim(...) is from the library "elim.lib",
      // it is loaded as a result of loading "general.lib"
      J=simplify(elim(J, E), 1);
      // get the expression of the next Chern class
      // in terms of the Chern classes of the initial vector bundle
      rez=rez+list( -subst(  J[1], x@, 0)  );
    }
  }
  else // otherwise compute only the needed Chern classes
  {
    C=symm(ss, M); // only the needed Chern classes
    int j;
    i=1;
    // the maximal number of optional parameters to be considered does not exceed N,
    // i.e., the rank of the symmetric power
    int NN = min( size(#), N);
    for(j=1; j <= NN; j++) // process the optional parameters
    {
      // process the optional parameters only untill they are not bigger than N;
      // notice they are positive anyway after integer_list(...)
      if( #[j]<=N  )
      {
        for( ; i<=#[j];i++)
        {
          if(i<=r)
          {
            // add the relevant formulas for the Chern classes in terms of the Chern roots
            I=I, c[i]-sym[i];
          }
        }
        J= I, x@-C[ #[j]];
        // Notice that elim(...) is from the library "elim.lib",
        // it is loaded as a result of loading "general.lib"
        J=simplify(elim(J, E), 1);
        // get the expression of the next Chern class
        // in terms of the Chern classes of the initial vector bundle
        rez=rez+list( -subst(  J[1], x@, 0)  );
      }
      else // get out from the loop
      {
        break;
      }
    }
  }
  // used because Singular seems not to be able to apply maps to empty lists (see below)
  if(size(rez)==0)
  {
    return(list(N, list()));
  }
  setring br@; // come back to the initial base ring
  // define the specialization homomorphism,
  // evaluate the formulas for the Chern classes on their given values
  execute( "map FF = r@,0,"+varstr(br@)+";" );
  list rez=FF( rez ); // bring the result back to the initial ring
  return( list( N, rez  ) ); // return the result together with the rank of the symmetric power
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (c(1..5)), dp;
  list l=c(1..5);
  // the rank and the Chern classes of the second symmetric power of a vector bundle of rank 3
  print( chSymm(2, 3, l) );
  // the rank and the first 3 Chern classes
  // of the second symmetric power of a vector bundle of rank 5
  print( chSymm(2, 5, l, 1, 2, 3) );
}
//----------------------------------------------------------------------------------

proc chSymm2L(int r, list c)
"USAGE:   chSymm2L(r, c); r integer, c list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern classes of the second symmetric power of a vector bundle
EXAMPLE:  example chSymm2L; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  // insure that the entries of c are polynomials
  // in order to be able to apply maps
  int i;
  for(i=1;i<=size(c);i++)
  {
    c[i]=poly(c[i]);
  }
  if(r<0) // if the rank is negative
  {
    print("The rank of a vector bundle can non be negative");
    return(list()); // return the empty list in this case
  }
  if(r==0) // if we deal with the zero bundle
  {
    return( list( 0, list() ) ); // return the data corresponding to the zero bundle
  }
  //-----------------------------------
  // from now on we are in the case r>0
  //-----------------------------------
  c=append_by_zeroes(r, c);
  c=c[1..r];
  def br@=basering; // remember the base ring
  // add  additional variables to the base ring
  execute("ring r@=(" + charstr(basering) +  "), ("+varstr(basering)+", t@, c@(1..r)), dp;" );
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c;
  for(i=1;i<=r;i++)
  {
    c[i]=c@(i)*t@^i;
  }
  poly f = chSymm2LP(r,c); // get the total Chern class using the formula of Lascoux
  matrix CF = coeffs(f, t@);
  int N=r*(r+1) div 2;
  list rez; // write the coefficients in front of the powers of t@ into a list
  for(i=1;i<=N;i++)
  {
    rez=rez+list(CF[i+1,1]);
  }
  setring br@; // come back to the initial base ring
  execute("map FF = r@,"+varstr(br@)+",0, c[1..r];"); // define the specialization homomorphism
  return( list(N, FF( rez )) ); // bring the result to the initial ring
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (c(1..2)), dp;
  list l=c(1..2);
  // the Chern classes of the second symmetric power of a vector bundle of rank 2
  print( chSymm2L(2, l));
}
//---------------------------------------------------------------------------------------

proc chSymm2LP(int r, list c)
"USAGE:   chSymm2LP(r, c); r integer, c list of polynomials
RETURN:   poly
PURPOSE:  computes the total Chern class of the second symmetric power of a vector bundle
EXAMPLE:  example chSymm2LP; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  if(r<0) // if the rank is negative
  {
    print("The rank of a vector bundle can non be negative");
    return(1); // return 1 in this case
  }
  if(r==0) // if we deal with the zero bundle
  {
    return( 1 ); // return 1 in this case
  }
  //-------------------------------------------
  // from now on we are in the case r > 0
  //-------------------------------------------
  c=append_by_zeroes(r, c);
  c=c[1..r];
  list I; // the partition (1,2,...,r) will be stored here
  int i;
  for(i=1;i<=r;i++)
  {
    I=I+list(i);
  }
  list PU = partUnder(I); // compute the partitions under I
  int sz=size(PU); // get their number
  poly rez; // the result will be computed here
  list J;
  poly cf;
  int ex;
  // implement the formula of Lascoux
  for(i=1;i<=sz;i++)
  {
    J=PU[i];
    ex=sum(J)- r*(r-1) div 2;
    if(ex>=0)
    {
      cf=bigint(2)^ex*IJcoef(I, J);
    }
    else
    {
      cf=IJcoef(I, J)/bigint(2)^(-ex);
    }
    rez = rez + cf * SchurCh(J, c );
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (c(1..2)), ws(1, 2);
  list l=c(1..2);
  // the total Chern class of the second symmetric power of a vector bundle of rank 2
  print( chSymm2LP(2, l));
}
//---------------------------------------------------------------------------------------

proc chWedge(int k, int r, list c, list #)
"USAGE:   chWedge(k, r, c [,pos]);  k, r integers, c list of polynomials, pos list of integers
RETURN:   list with entries: int N, list of polynomials l
PURPOSE:  computes the rank and the Chern classes of the exterior power of a vector bundle
EXAMPLE:  example chWedge; shows an example
NOTE:     for the second exterior power chWedge2L(...) could be faster
"
{
  // insure that the entries of c are polynomials
  // in order to be able to apply maps
  int i;
  for(i=1;i<=size(c);i++)
  {
    c[i]=poly(c[i]);
  }
  if(r<0) // if the rank is negative
  {
    print("The rank of a vector bundle can non be negative");
    return(list()); // return the empty list in this case
  }
  if(r==0) // if we deal with the zero bundle
  {
    return( list( 0, list() ) ); // return the data corresponding to the zero bundle
  }
  //-------------------------------------------
  // from now on we are in the case r > 0
  //-------------------------------------------
  if(k<0)
  {
    print("You are trying to compute a negative exterior power of a vector bundle");
    return( list(0, list() ) ); // assume such a power to be just a zero bundle
  }
  if(k==0) // the zeroth exterior power is the trivial line bundle
  {
    return( list(1, list(0)) );
  }
  if(k==1) // the first exterior power is equal to the vector bundle itself
  {
    c=append_by_zeroes(r, c);
    c=c[1..r];
    return(list(r, c));
  }
  //---------------------------------------
  // from now on we are in the case k > 2
  //---------------------------------------
  // if the length of the list of Chern classes is smaller than the rank of the vector bundle,
  // the higher classes are assumed to be zero and the list is appended by zeroes up to length r
  c=append_by_zeroes(r, c);
  // if the length of the list of the Chern classes is greater than the rank
    c=c[1..r]; // throw away the redundant data
  //------------------------------------------
  // from now on the length of c is r > 0
  //------------------------------------------
  if( k>r ) // if k>r, the exterior power is zero
  {
    return( list( int(0), list()  ) );
  }
  //-----------------------------------------------
  // from now on we are in the case 0 < k <= r = n
  //-----------------------------------------------
  if(k==r)
  {
    return(list( int(1), list( c(1) ) ) );
  }
  //-----------------------------------------------
  // from now on we are in the case 0 < k < r = n
  //-----------------------------------------------
  list LM = integer_list(#);
  int M=LM[2]; // maximum among the optional parameters if there are any, zero otherwise
  # = LM[1]; // take into account only the first integer optional parameters that are positive
  //-----------------------------
  // Let us compute now
  //-----------------------------
  def br@=basering; // remember the base ring
  // add additional variables a@(1..r), x@ to the base ring
  execute("ring r@= (" + charstr(basering) + "), (x@,"+varstr(basering)+", a@(1..r)), lp;" );
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c = F(c); // embed c into the bigger ring
  list rez; // the result should be computed here
  poly E = product( list( a@(1..r ) )  ); // product of the Chern roots to be eliminaned
  list ss=ChernRootsWedge(k,  list( a@(1..r) )); // list of the Chern roots of the exterior product
  int N=size(ss); // length of ss, equals the rank of the exterior product
  // list of the Chern classes of the initial vector bundle in terms of their Chern roots
  list sym=symm(list(a@(1..r)));
  // the entries here will be the Chern classes we need
  // expressed in  terms of the Chern roots of the initial vector bundle
  list C;
  ideal I, J;
  if( size(#) == 0 ) // if there are no optional parameters, compute all Chern classes
  {
    // the entries here are the Chern classes we need
    // expressed in  terms of the Chern roots of the initial vector bundle
    C=symm(ss);
    for(i=1;i<=N;i++) // eliminate the Chern roots
    {
      if(i<= r) // first add all relevant  formulas for the Chern classes in terms of Chern roots
      {
        I=I, c[i]-sym[i];
      }
      J = I,  x@-C[i];
      // Notice that elim(...) is from the library "elim.lib",
      // it is loaded as a result of loading "general.lib"
      J=simplify(elim(J, E), 1);
      // get the expression of the next Chern class
      // in terms of the Chern classes of the initial vector bundle
      rez=rez+list( -subst(  J[1], x@, 0)  );
    }
  }
  else // otherwise compute only the needed Chern classes
  {
    // the entries here are the Chern classes we need
    // expressed in  terms of the Chern roots of the initial vector bundle
    C=symm(ss, M);
    int j;
    i=1;
    // the maximal number of optional parameters to be considered
    // does not exceed N, the rank of the exterior power
    int NN = min( size(#), N);
    for(j=1; j <= NN; j++) // process the optional parameters
    {
      // process the optional parameters only untill they are not bigger than N;
      // notice they are positive anyway after integer_list(...)
      if( #[j]<=N  )
      {
        for( ; i<=#[j]; i++)
        {
          if( i<=r )
          {
            // add the relevant formulas for the Chern classes in terms of the Chern roots
            I=I, c[i]-sym[i];
          }
        }
        J= I, x@-C[ #[j]];
        // Notice that elim(...) is from the library "elim.lib",
        // it is loaded as a result of loading "general.lib"
        J=simplify(elim(J, E), 1);
        // get the expression of the next Chern class
        // in terms of the Chern classes of the initial vector bundle
        rez=rez+list( -subst(  J[1], x@, 0)  );
      }
      else // get out from the loop
      {
        break;
      }
    }
  }
  // used because Singular seems not to be able to apply maps to empty lists (see below)
  if(size(rez)==0)
  {
    return(list(N, list()));
  }
  setring br@; // come back to the initial base ring
  // define the specialization homomorphism,
  // evaluate the formulas for the Chern classes on their given values
  execute( "map FF = r@,0,"+varstr(br@)+";" );
  list rez=FF( rez ); // bring the result back to the initial ring
  return( list( N, rez  ) ); //return the rank and the Chern classes of the exterior product
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (c(1..5)), dp;
  list l=c(1..5);
  // the rank and the Chern classes of the second exterior power of a vector bundle of rank 3
  print( chWedge(2, 3, l) );
  // the rank and the first 3 Chern classes
  // of the fourth exterior power of a vector bundle of rank 5
  print( chWedge(4, 5, l, 1, 2, 3) );
}
//---------------------------------------------------------------------------------

proc chWedge2L(int r, list c)
"USAGE:   chWedge2L(r, c );  r integer, c list of polynomials
RETURN:   list of polynomials
PURPOSE:  computes the Chern classes of the second exterior power of a vector bundle
EXAMPLE:  example chWedge2L; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  // insure that the entries of c are polynomials
  // in order to be able to apply maps
  int i;
  for(i=1;i<=size(c);i++)
  {
    c[i]=poly(c[i]);
  }
  if(r<0) // if the rank is negative
  {
    print("The rank of a vector bundle can non be negative");
    return(list()); // return the empty list in this case
  }
  if(r==0) // if we deal with the zero bundle
  {
    return( list( 0, list() ) ); // return the data corresponding to the zero bundle
  }
  //-------------------------------------------
  // from now on we are in the case r > 0
  //-------------------------------------------
  c=append_by_zeroes(r, c);
  c=c[1..r];
  def br@=basering; // remember the base ring
  // add  additional variables to the base ring
  execute("ring r@=(" + charstr(basering) +  "), ("+varstr(basering)+", t@, c@(1..r)), dp;" );
  execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
  list c;
  for(i=1;i<=r;i++)
  {
    c[i]=c@(i)*t@^i;
  }
  poly f = chWedge2LP(r,c); // get the total Chern class using the formula of Lascoux
  matrix CF = coeffs(f, t@);
  int N=r*(r-1) div 2;
  list rez; // write its coefficients in front of the powers of t@ to a list
  for(i=1;i<=N;i++)
  {
    rez=rez+list(CF[i+1,1]);
  }
  setring br@; // come back to the initial base ring
  execute("map FF = r@,"+varstr(br@)+",0, c[1..r];"); // define the specialization homomorphism
  return( list(N, FF( rez )) ); // bring the result to the initial ring
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (c(1..3)), dp;
  list l=c(1..3);
  // the Chern classes of the second exterior power of a vector bundle of rank 3
  print(chWedge2L(3, l));
}
//---------------------------------------------------------------------------------------

proc chWedge2LP(int r, list c)
"USAGE:   chWedge2LP(r, c );  r integer, c list of polynomials
RETURN:   poly
PURPOSE:  computes the total Chern class of the second exterior power of a vector bundle
EXAMPLE:  example chWedge2LP; shows an example
NOTE:     Implementation of the formula of Lascoux, the Schur polynomials are computed
          using the second Jacobi-Trudi formula (in terms of the Chern classes)
"
{
  if(r<0) // if the rank is negative
  {
    print("The rank of a vector bundle can non be negative");
    return(1); // return 1 in this case
  }
  if(r==0) // if we deal with the zero bundle
  {
    return( 1 ); // return 1 in this case
  }
  //-------------------------------------------
  // from now on we are in the case r > 0
  //-------------------------------------------
  c=append_by_zeroes(r, c);
  c=c[1..r];
  list I; // the partition (0,1,...,r-1) will be stored here
  int i;
  for(i=0;i<=r-1;i++)
  {
    I=I+list(i);
  }
  list PU = partUnder(I); // compute the partitions under I
  int sz=size(PU); // get their number
  poly rez; // the result will be computed here
  list J;
  poly cf;
  // implement the Lascoux formula
  for(i=1;i<=sz;i++)
  {
    J=PU[i];
    cf = IJcoef(I,J)/bigint(2)^( r*(r-1) div 2-sum(J)  );
    rez = rez + cf * SchurCh(J, c );
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (c(1..3)), ws(1,2,3);
  list l=c(1..3);
  // the total Chern class of the second exterior power of a vector bundle of rank 3
  print(chWedge2LP(3, l));
}
//---------------------------------------------------------------------------------------

proc todd(list c, list #)
"USAGE:   todd(l [, n] );  l a list of polynomials, n integer
RETURN:   list of polynomials
PURPOSE:  computes [the first n] terms of the Todd class
EXAMPLE:  example todd; shows an example
NOTE:     returns an empty list if l is empty
"
{
  int i, j, k;
  // insure that the entries of c are polynomials
  // in order to be able to apply maps
  for(i=1;i<=size(c); i++)
  {
    c[i]=poly(c[i]);
  }
  int n;
  # = integer_list(#)[1]; // take into account only the first integer entries that are positive
  if( size(#) == 0 ) // if there are no  optional parameters
  {
    n = size(c);
  }
  else
  {
    // set n to be 0, if the parameter is non-positive,
    // set n to the value of the parameter otherwise
    n = max( #[1], 0 );
    c = append_by_zeroes(n, c); // append c by zeroes if the length of c  is smaller than n
    if(n!=0) // throw away the redundant data if n is positive and smaller than the length of c
    {
      c = c[1..n];
    }
  }
  if(n==0) // return the empty list
  {
    return(list());
  }
  else // otherwise proceed as follows
  {
    def br@=basering; // remember the base ring
    // add  additional variables to the base ring
    execute("ring r@=(" + charstr(basering) +  "), ("+varstr(basering)+", a@, c@(1..n)), dp;" );
    execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
    list c=F(c); // embed c into the bigger ring
    list prev;
    list next;
    next=tdTerms(n, c@(1)); // the Todd class terms of a line budle
    list step = tdTerms(n, a@);
    poly f;
    list hC=c@(1)-a@; // "old" first Chern class
    for(k=2;k<=n;k++) // do n-1 iterations
    {
      prev=next;
      next=list();
      hC=hC+list( c@(k)-a@*hC[k-1] ); // "old" k-th Chern class
      for(i=0;i<k;i++) // these terms have already been computed in the previous iterations
      {
        next = next + list(prev[i+1]);
      }
      for(i=k;i<=n;i++) // new values in terms of "old" Chern classes and the Chern root a
      {
        f=0;
        for(j=0; j<=i; j++)
        {
          f=f + step[j+1]*prev[i-j+1];
        }
        // substitute the old values of Chern classes
        // by their expressions in the new ones and the Chern root a
        for(j=1;j<k;j++)
        {
          f=subst(f, c@(j), hC[j] );
        }
        f=reduce(f, std(hC[k]) ); // eliminate the Chern root
        next = next + list(f);
      }
    }
    next = delete(next, 1); // throw away the zeroth term which is always equal to 1
    setring br@; // come back to the initial base ring
    execute("map FF = r@,"+varstr(br@)+",0, c[1..n];"); // define the specialization homomorphism
    return( FF( next ) ); // bring the result to the initial ring
  }
}
example
{
  "EXAMPLE:";echo =2;
  // the terms of the Todd class up to degree 5
  // in terms of the Chern classes c(1), c(2), c(3), c(4), c(5)
  ring r=0, (c(1..5)), dp;
  list l=c(1..5);
  print( todd( l ) );

  // in the same situation compute only first two terms
  print( todd(l, 2) );

  // compute the first 5 terms corresponding to the Chern clases c(1), c(2)
  l=c(1..2);
  print( todd(l, 5) );
}
//------------------------------------------------------------------------------------------

proc toddE(list c)
"USAGE:   toddE(l);  l a list of polynomials
RETURN:   polynomial
PURPOSE:  computes the highest relevant term of the Todd class
EXAMPLE:  example toddE; shows an example
NOTE:     returns an empty list if l is empty,
          very inefficient because the elimination is used, included for comparison with todd(c)
"
{
  int i;
  for(i=1;i<=size(c);i++)
  {
    c[i]=poly( c[i] );
  }
  int n=size(c);
  if(n==0) // return the empty list if c is empty
  {
    return(list());
  }
  else
  {
    def br@=basering; // remember the base ring
    // add additional variables a@(1..n), x@ to the base ring
    execute("ring r@=(" + charstr(basering) + "), (x@,"+varstr(basering)+", a@(1..n)), dp;" );
    execute( "map F= br@,"+varstr(br@)+";" ); // define the corresponding inclusion of rings
    list c=F(c); // embed c into the bigger ring
    int j;
    int k;
    poly E = a@(1); // to be the product of the Chern roots that will be eliminated later
    list ss=tdTerms( n, a@(1) );
    list next; // to be the terms of the Todd class corresponding to the next Chern root
    for(i=2;i<=n;i++) // compute the terms of the Todd class in terms of the Chern roots
    {
      E=E*a@(i); // to compute the product of variables to be eliminated
      next=tdTerms( n, a@(i) );
      for(j=n;j>=1;j--)
      {
        for(k=0;k<j;k++)
        {
          ss[j+1]=ss[j+1]+ss[k+1]*next[j-k+1];
        }
      }
    }
    ideal I=x@ - ss[n+1]; // formula for the highest degree term of the Todd class
    list sym=symm(list(a@(1..n))); // expressions for the Chern classes in terms of the Chern roots
    for(i=1;i<=n;i++)
    {
      I=I, c[i]-sym[i]; // add the relations
    }
    I=simplify(elim(I, E), 1); // eliminate the Chern roots
    poly rez=-subst(I[1],x@, 0); // get the required formula
    setring br@; // come back to the initial base ring
    // define the specialization homomorphism (all added variables are set to zero)
    execute( "map FF = r@,0, "+varstr(br@)+";" );
    poly rez=FF( rez ); // bring the result back to the initial base ring
    return(rez);
  }
}
example
{
  "EXAMPLE:";echo =2;
  // first  3 terms of the Todd class in terms of the Chern classes c(1), c(2), c(3)
  ring r=0, (c(1..3)), dp;
  list l;
  //first term
  l=c(1);
  print( toddE( l ) );
  // second term
  l=c(1..2);
  print( toddE( l ) );
  // third term
  l=c(1..3);
  print( toddE( l ) );
}
//---------------------------------------------------------------------------------

proc  Bern(int n)
"USAGE:   Bern(n);  n non-negative integer
RETURN:   list of numbers
PURPOSE:  computes the list of (second) Bernoulli numbers from B(0) to B(n)
EXAMPLE:  example Bern; shows an example
NOTE:     needs a base ring to be defined, returns an empty list if n is negative,
          uses the Akiyama-Tanigawa algorithm
"
{
  // the Akiyama-Tanigawa algorithm
  //could be replaced by a more efficient one
  list rez, steprez;
  int i, j;
  if(n<0) // if n is negative, return the empty list
  {
    return(list());
  }
  for(i=0;i<=n;i++)
  {
    steprez=steprez+list( 1/number(i+1) );
    for(j=i;j>=1;j--)
    {
      steprez[j]=j*(steprez[j]-steprez[j+1]);
    }
    rez=rez+list(steprez[1]);
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  // first 10 Bernoulli numbers: B(0), ..., B(9)
  ring r=0,(t), dp;
  print( Bern(9) );
}
//---------------------------------------------------------------------------------

proc tdCf(int n)
"USAGE:   tdCf(n);  n integer
RETURN:   list of rational numbers
PURPOSE:  computes up to degree n the coefficients of the Todd class of a line bundle
EXAMPLE:  example tdCf; shows an example
NOTE:
"
{
  list rez=Bern(n); // notice that Bern(n) is able to take care of negative n
  int i;
  for(i=1;i<=n+1;i++)
  {
  rez[i]=rez[i]/factorial(i-1);
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  // first 5 coefficients
  ring r=0,(t), dp;
  print( tdCf(4) );
}
//---------------------------------------------------------------------------------

proc tdTerms(int n, poly f)
"USAGE:   tdTerms(n, f);  n integer, f polynomial
RETURN:   list of polynomials
PURPOSE:  computes the terms of the Todd class of the line bundle with the Chern root f
EXAMPLE:  example tdTerms; shows an example
NOTE:
"
{
  list rez=Bern(n); // notice that Bern(n) takes care of negative n
  int i;
  for(i=1;i<=n+1;i++)
  {
  rez[i]=( rez[i]/factorial(i-1) )* f^(i-1);
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (t), ls;;
  // the terms of the Todd class of a line bundle with Chern root t up to degree 4
  print( tdTerms(4, t) );
}
//---------------------------------------------------------------------------------

proc tdFactor(int n, poly t)
"USAGE:   tdFactor(n, a);  n integer, a polynomial
RETURN:   polynomial
PURPOSE:  computes up to degree n the Todd class
          of the line bundle coresponding to the Chern root t
EXAMPLE:  example tdFactor; shows an example
NOTE:     returns 0 if n is negative
"
{
  int i;
  poly rez=0;
  list l=Bern(n); // get the coefficients
  for(i=0; i<=n; i++) // form the polynomial
  {
    rez=rez+(l[i+1]/factorial(i))*t^i;
  }
  return(rez);
}
example
{
  "EXAMPLE:";echo =2;
  // the Todd class up do degree 4
  ring r=0,(t), ls;
  print( tdFactor(4, t) );
}
//---------------------------------------------------------------------------------

proc cProj(int n)
"USAGE:   cProj(n);  n  integer
RETURN:   list of integers
PURPOSE:  computes the terms of positive degree of the total Chern class
          of the tangent bundle on the complex projective space
EXAMPLE:  example cProj; shows an example
NOTE:
"
{
  if(n<0)
  {
    print("The dimension of the projective space must be non-negative!");
    return(list()); // return the empty list in this case
  }
  else
  {
    list rez;
    int i;
    for(i=1;i<=n;i++)
    {
      rez=rez+list( binomial(n+1, i) );
    }
    return(rez);
  }
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (t), dp;
  // the coefficients of the total Chern class of the complex projective line
  print( cProj(1) );

  // the coefficients of the total Chern class of the complex projective line
  print( cProj(2) );

  // the coefficients of the total Chern class of the complex projective line
  print( cProj(3) );
}
//------------------------------------------------------------------------------------------

proc chProj(int n)
"USAGE:   chProj(n);  n  integer
RETURN:   list of (rational) numbers
PURPOSE:  computes the terms of the Chern character of the tangent bundle
          on the complex projective space
EXAMPLE:  example chProj; shows an example
NOTE:
"
{
  if(n<0)
  {
    print("The dimension of the projective space must be non-negative!");
    return( list() ); // return the empty list in this case
  }
  else
  {
    list rez=list(number(n));
    int i;
    for(i=1;i<=n;i++)
    {
      rez=rez+list( (n+1)/factorial(i) );
    }
    return(rez);
  }
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (t), dp;
  // the coefficients of the Chern character of the complex projective line
  print( chProj(1) );

  // the coefficients of the Chern character of the complex projective plane
  print( chProj(2) );

  // the coefficients of the Chern character of the complex 3-dimentional projectice space
  print( chProj(3) );
}
//------------------------------------------------------------------------------------------

proc tdProj(int n)
"USAGE:   tdProj(n);  n  integer
RETURN:   list of (rational) numbers
PURPOSE:  computes the terms of the Todd class
          of the (tangent bundle of the) complex projective space
EXAMPLE:  example tdProj; shows an example
NOTE:
"
{
  if(n<0)
  {
    print("The dimension of the projective space must be non-negative!");
    return( list() ); // return the empty list in this case
  }
  else
  {
    def br@=basering; // remember the base ring
    ring r@= 0, t@, lp; // ring with one variable t@
    ideal T=std( t@^(n+1) );
    poly f= tdFactor(n, t@);
    f=reduce( f^(n+1), T);
    matrix C = coeffs(f, t@);
    list rez;
    int i;
    for(i=0;i<=n;i++)
    {
     rez=rez+list(C[i+1, 1]);
    }
    setring br@; // come back to the initial base ring
    map FF= r@, 0 ; // define the specialization homomorphism t@=0
    return(FF(rez)); // bring the result to the base ring
  }
}
example
{
  "EXAMPLE:";echo =2;
  ring r=0, (t), dp;

  // the coefficients of the Todd class of the complex projective line
  print( tdProj(1) );

  // the coefficients of the Todd class of the complex projective line
  print( tdProj(2) );

  // the coefficients of the Todd class of the complex projective line
  print( tdProj(3) );
}
//------------------------------------------------------------------------------------------

proc eulerChProj(int n, def r, list c)
"USAGE:   eulerChProj(n, r, c);  n  integer, r polynomial (or integer), c list of polynomials
RETURN:   polynomial
PURPOSE:  computes the Euler characteristic of a vector bundle on P_n
          in terms of its rank and Chern classses
EXAMPLE:  example eulerChProj; shows an example
NOTE:
"
{
  if(n<0)
  {
    print("The dimension of the projective space must be non-negative!");
    return(0); // return zero in this case
  }
  else
  {
    if(n==0)
    {
      return(r);
    }
    // now n is at least 1
    c=append_by_zeroes(n, c); // append c by zeroes if its size is smaller than n
    c=c[1..n]; // throw away the redundant data
    // now the size of c is n
    list td = tdProj(n); // terms of the Todd class of P_n
    list ch = list(r) + chAll(c); // terms of the Chern character of the vector bundle
    return( rHRR(n, ch, td) );
  }
}
example
{
  "EXAMPLE:";echo =2;
  ring h=0, (r, c(1..3)),  ws(0,1,2,3);
  list l=c(1..3);
  // the Euler characteristic of a vector bundle on the projective line
  print( eulerChProj(1, r, l) );

  // the Euler characteristic of a vector bundle on the projective plane
  print( eulerChProj(2, r, l) );

  // the Euler characteristic of a vector bundle on P_3
  print( eulerChProj(3, r, l) );

  // assume now that we have a bundle framed at a subplane of P_3
  // this implies c(1)=c(2)=0
  l= 0, 0, c(3);

  // the Euler characteristic is
  print( eulerChProj(3, r, l) );
  // which implies that c(3) must be even in this case
}
//-------------------------------------------------------

proc chNumbersProj(int n)
"USAGE:   chNumbersProj(n);  n  integer
RETURN:   list of integers
PURPOSE:  computes the Chern numbers of the projective space P_n
EXAMPLE:  example chNumbersProj; shows an example
NOTE:
"
{
  return( chNumbers( n, cProj(n) ) );
}
example
{
  "EXAMPLE:";echo =2;
  ring h=0, (t), dp;
  // The Chern numbers of the projective plane P_2:
  print( chNumbersProj(2) );

  // The Chern numbers of P_3:
  print( chNumbersProj(3) );
}
//-------------------------------------------------------

proc classpoly(list l, poly t)
"USAGE:   classpoly(l, t);   l list of polynomials, t polynomial
RETURN:   polynomial
PURPOSE:  computes the polynomial in t with coefficients being the entries of l
EXAMPLE:  example classpoly; shows an example
NOTE:
"
{
  int n=size(l);
  poly pow=1; // powers of t will be compured here
  poly rez=0; // result will be computed here
  int i;
  for(i=1; i<=n; i++)
  {
    pow=pow*t; // compute the required power of t
    // add the i-th entry of l multiplied by the corresponding power of t to the result
    rez=rez + l[i]*pow;
  }
  return( rez );
}
example
{
  "EXAMPLE:";echo=2;
  ring r=0, (c(1..5), t), ds;
  list l=c(1..5);
  // get the polynomial c(1)*t + c(2)*t^2 + ... + c(5)*t^5
  print( classpoly(l, t) );
}
//----------------------------------------------------------------------------------------

proc chernPoly(list c, poly t)
"USAGE:   chernPoly(c, t);   c list of polynomials, t polynomial
RETURN:   polynomial
PURPOSE:  computes the Chern polynomial in t
EXAMPLE:  example chernPoly; shows an example
NOTE: does the same as toddPoly(...)
"
{
  return( 1+classpoly(c, t) );
}
example
{
  "EXAMPLE:";echo=2;
  ring r=0, (c(1..5), t), ds;
  list l=c(1..5);
  // get the Chern polynomial 1 + c(1)*t + c(2)*t^2 + ... + c(5)*t^5
  print( chernPoly(l, t) );
}
//----------------------------------------------------------------------------------------

proc chernCharPoly(poly r, list ch, poly t)
"USAGE:   chernCharPoly(r, ch, t);   r polynomial, ch list of polynomials, t polynomial
RETURN:   polynomial
PURPOSE:  computes the polynomial in t corresponding to the Chern character
EXAMPLE:  example chernpoly; shows an example
NOTE:
"
{
  return( r+classpoly(ch, t) );
}
example
{
  "EXAMPLE:";echo=2;
  ring h=0, (r, ch(1..5), t), ds;
  list l=ch(1..5);
  // get the polynomial r + ch(1)*t + ch(2)*t^2 + ... + ch(5)*t^5
  print( chernCharPoly(r, l, t) );
}
//----------------------------------------------------------------------------------------

proc toddPoly(list td, poly t)
"USAGE:   toddPoly(td, t);   td list of polynomials, t polynomial
RETURN:   polynomial
PURPOSE:  computes the polynomial in t corresponding to the Todd class
EXAMPLE:  example toddPoly; shows an example
NOTE:     does the same as chernPoly(...)
"
{
  return( 1+classpoly(td, t) );
}
example
{
  "EXAMPLE:"; echo=2;
  ring r=0, (td(1..5), c(1..5), t), ds;
  list l=td(1..5);
  // get the polynomial 1 + td(1)*t + td(2)*t^2 + ... + td(5)*t^5
  print( toddPoly(l, t) );
}
//---------------------------------------------------------------------------------------

proc rHRR(int N, list ch, list td)
"USAGE:   rHRR( N, ch, td);  N integer, ch, td  lists of polynomials
RETURN:   polynomial
PURPOSE:  computes the the main ingredient of the right-hand side
          of the Hirzebruch-Riemann-Roch formula
EXAMPLE:  example rHRR; shows an example
NOTE:     in order to get the right-hand side of the HRR formula
          one needs to be able to compute the degree of the output of this procedure
"
{
  poly rez; // to be the result
  int i;
  int nch=size(ch); // length of ch
  int ntd=size(td); // length of td
  for(i=1; i<=N+1; i++) // compute the highest degree term of ch.td
  {
    if( (i<=nch) && (N-i+2 <= ntd) )
    {
      rez = rez + ch[i]*td[N-i+2];
    }
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo=2;
  ring r=0, (td(0..3), ch(0..3)), dp;
  // Let ch(0), ch(1), ch(2), ch(3) be the terms of the Chern character
  // of a vector bundle E on a 3-fold X.
  list c = ch(0..3);
  // Let td(0), td(1), td(2), td(3) be the terms of the Todd class of X.
  list t = td(0..3);
  // Then the highest term of the product ch(E).td(X) is:
  print( rHRR(3, c, t) );
}
//---------------------------------------------------------------------------------------

proc SchurS(list I, list S)
"USAGE:   SchurS(I, S);  I list of integers representing a partition, S list of polynomials
RETURN:   poly
PURPOSE:  computes the Schur polynomial in the Segre classes S (of the dual vector bundle),
          i.e., in the  complete homogeneous symmetric polynomials, with respect to the partition I
EXAMPLE:  example SchurS; shows an example
NOTE:     if S are the Segre classes of the tautological bundle on a grassmanian,
          this gives the cohomology class of a Schubert cycle
"
{
  int m=size(I); // size of I
  S=list(1)+S; // add the zeroth Segre class
  int szS=size(S); // size of S
  int h,k;
  int in; // variable for the index of the required Segre class
  // construct the required m x m matrix from the first determinantal (Jacobi-Trudi) formula
  matrix M[m][m];
  for(h=1;h<=m;h++)
  {
    for(k=1;k<=m;k++)
    {
      in=I[k]+k-h; // compute the index
      if(in<0) // if it is negative, assume the corresponding Segre class to be zero
      {
        M[h,k]=0;
      }
      else
      {
        if(in>=szS) // if it is bigger than the number of the highest available Segre class in S
        {
          M[h, k]=0; // assume the corresponding Segre class is zero
        }
        else // otherwise
        {
          M[h, k]= S[in+1]; // use a value from S for the corresponding Segre class
        }
      }
    }
  }
  return(det(M)); // return the determinant of the computed matrix
}
example
{
  "EXAMPLE:"; echo=2;
  // The Schur polynomial corresponding to the partition 1,2,4
  // and the Segre classes 1, s(1), s(2),..., s(6)
  ring r=0,(s(1..6)), dp;
  list I=1,2,4;
  list S=s(1..6);
  print( SchurS(I, S) );
  // compare this with the Schur polynomial computed using Chern classes
  list C=chDual(chern(S));
  print( SchurCh(I, C) );
}
//---------------------------------------------------------------------------------------

proc SchurCh(list I, list C)
"USAGE:   SchurCh(I, C);  I list of integers representing a partition, C list of polynomials
RETURN:   poly
PURPOSE:  computes the Schur polynomial in the Chern classes C,
          i.e., in the elementary symmetric polynomials, with respect to the partition I
EXAMPLE:  example SchurCh; shows an example
NOTE:     if C are the Chern classes of the tautological bundle on a grassmanian,
          this gives the cohomology class of a Schubert cycle
"
{
  I=dualPart(I); // dual partition to I
  int m=size(I); // size of I
  C=list(1)+C; // add the zeroth Chern class
  int szC=size(C); // size of C
  int h,k;
  int in; // variable for the index of the required Chern class
  // construct the required m x m matrix from the second determinantal (Jacobi-Trudi) formula
  matrix M[m][m];
  for(h=1;h<=m;h++)
  {
    for(k=1;k<=m;k++)
    {
      in=I[k]+k-h; // compute the index
      if(in<0) // if it is negative, assume the corresponding Chern class to be zero
      {
        M[h,k]=0;
      }
      else
      {
        if(in>=szC) // if it is bigger than the number of the highest available Chern class in C
        {
          M[h, k]=0; // assume the corresponding Chern class is zero
        }
        else // otherwise
        {
          M[h, k]= C[in+1]; // use a value from C for the corresponding Chern class
        }
      }
    }
  }
  return(det(M)); // return the determinant of the computed matrix
}
example
{
  "EXAMPLE:"; echo=2;
  // The Schur polynomial corresponding to the partition 1,2,4
  // and the Chern classes c(1), c(2), c(3)
  ring r=0,(c(1..3)), dp;
  list I=1,2,4;
  list C=c(1..3);
  print( SchurCh(I, C) );
  // Compare this with the Schur polynomial computed using Segre classes
  list S=segre( chDual( list(c(1..3)) ), 6 );
  print(SchurS(I,S));
}
//---------------------------------------------------------------------------------------

proc part(int m, int n)
"USAGE:   part( m, n );  m positive integer, n non-negative integer
RETURN:   list of lists
PURPOSE:  computes all partitions of integers not exceeding n into m non-negative summands
EXAMPLE:  example part; shows an example
NOTE:     if n is negative or m is non-positive, the list with one empty entry is returned
"
{
  if( n<0 ) // if n is negative
  {
    return(list(list())); // return the list with one empty entry
  }
  if(n==0) // if n equals 0, there is only one partition of 0 into m non-negative summands
  {
    return(list(listSame(0,m))); // return the list with one entry consistion of m zeroes
  }
  // otherwise proceed recursively
  list rez=part(m, n-1); // get all partitions for n-1
  int i;
  for(i=1;i<=m;i++) // for every i between 1 and m, add the partitions with exactly
  {
    rez=rez + appendToAll( part(m-i, n-1), listSame(n, i) ); // i summands equal to n
  }
  return(rez); // return the result
}
example
{
  "EXAMPLE:"; echo=2;
  // partitions into 3 summands of numbers not exceeding 1
  print( part(3, 1) );
}
//---------------------------------------------------------------------------------------

proc dualPart(list I, list #)
"USAGE:   dualPart( I [,N] );  I list of integers, N integer
RETURN:   list of integers
PURPOSE:  computes the partition dual (conjugate) to I
EXAMPLE:  example dualPart; shows an example
NOTE:     the result is extended by zeroes to length N if an optional integer
          parameter N is given and the length of the computed  dual partition
          is smaller than N
"
{
  int m= size(I); // size of I
  if(m==0) // if I is the empty list
  {
    print("You are trying to compute the dual of the empty partition!");
    print("The partition with one zero is returned.");
    return(list(0));
  }
  // compute the dual partition
  list J; // the result will be computed here
  int i;
  int j=I[1];
  int k;
  for(k=1;k<=j;k++)
  {
    J=list(m)+J;
  }
  for(i=2;i<=m;i++)
  {
    j=I[i]-I[i-1];
    for(k=1;k<=j;k++)
    {
      J=list(m-i+1)+J;
    }
  }
  if(size(J)==0) // if the dual partition J is empty (if I consists of zeroes)
  {
    J = list(0); // add zero to the result
  }
  if(size(#)>0) // if there is an optional parameter N
  {
    if( is_integer( #[1] ) ) // if the parameter is an integer,
    {
      if( size(J) < #[1] ) // if N  is bigger than the length of J,
      {
        J=listSame(0, #[1]-size(J))+J; // extend J by zeroes to length N
      }
    }
  }
  return(J); // return the result
}
example
{
  "EXAMPLE:"; echo =2;
  // dual partition to (1, 3, 4):
  list I = 1, 3, 4;
  print( dualPart(I) );
}
//---------------------------------------------------------------------------------------

proc PartC(list I, int m)
"USAGE:   PartC( I, m);  I list of integers, m integer
RETURN:   list of integers
PURPOSE:  commputes the complement of a partition with respect to m
EXAMPLE:  example PartC; shows an example
NOTE:     returns the zero partition if the maximal element of the partition is smaller than m
"
{
  int n=size(I); // size of I
  if( m<I[n] ) // if m is smaller than the last term of I,
  {
    // give a warning
    print("You are trying to compute a complement of a partition with respect");
    print("to a number that is smaller than the maximal summand of the partition!");
    print("The zero partition is returned.");
    return(list(0)); // and return the zero partition
  }
  list J; // the result will be computed here
  int i;
  for(i=n;i>=1;i--) // invert the order of numbers
  {
    J=J+list(m-I[i]); // and substitute them by their complemenst to m
  }
  return(J); // return the result
}
example
{
  "EXAMPLE:"; echo =2;
  // Complement of the partition (1, 3, 4) with respect to 5
  list I = 1, 3, 4;
  print( PartC(I, 5) );
}
//---------------------------------------------------------------------------------------

proc partOver(int n, list J)
"USAGE:   partOver( n, J);  n integer, J list of integers (partition)
RETURN:   list of lists
PURPOSE:  computes the partitions over a given one  with summands not exceeding n
EXAMPLE:  example partOver; shows an example
NOTE:
"
{
  int m=size(J); // size of J
  if( m==0 ) // if J is an empty list
  {
    // give a warning
    print("You are trying to compute partitions over an empty partition!");
    return( list() ); // and return the emty list
  }
  if( J[m] > n ) // if the biggest summand of the partition is bigger than n
  {
    return( list( ) ); // return the emty list
  }
  if( J[m] == 0 ) // if J consists of zeroes
  {
    return( part(m,n) ); // return all partitions of n into m summands
  }
  // now J is non-empty, contains con-zero summands, has partitions over it
  list rez1; // the result will be computed here
  int i,j;
  if(m==1) // if J has only one element
  {
    for(j=J[1]; j<=n; j++) // run through the integers from J[1] to n
    {
      rez1=rez1 + list(j); // add the corresponding one element lists to the result
    }
    return(rez1); // return the result
  }
  // now J has at least two elements
  // get the partitions over the partition without the last summand
  list rez = partOver(n, delete(J, m));
  int sz=size(rez); // number of such partitions
  list P;
  int last;
  for(i=1; i<=sz; i++) // run trough all such partitions
  {
    P=rez[i]; // for each partition P of this type
    last = max( P[size(P)], J[m] );
    for(j = last;j<= n;j++) // run through the integers exceding the last summands of P and J
    {
      // append them to P at the end and add the resulting partition to the result
      rez1=rez1 + list(P+list(j));
    }
  }
  return(rez1); // return the result
}
example
{
  "EXAMPLE:"; echo =2;
  // Partitions over the partition (3, 3, 4) with summands not exceeding 4
  list I = 3, 3, 4;
  print( partOver(4, I) );
}
//---------------------------------------------------------------------------------------

proc partUnder(list J)
"USAGE:   partUnder(J);  J list of integers (partition)
RETURN:   list of lists
PURPOSE:  computes the partitions under a given one
EXAMPLE:  example partUnder; shows an example
NOTE:
"
{
  int m=size(J); // size of J
  if(m==0) // if J is empty
  {
    return(list()); // return an empty list
  }
  list rez1; // the result will be computed here
  int i;
  if(m==1) // if J contains only one element
  {
    for(i=0; i<=J[1]; i++)
    {
      rez1=rez1+list(list(i));
    }
  }
  // now J contains at least two elements
  list rez;
  int Jlast=J[m]; // last element of J
  rez = partUnder(delete(J, m)); // partitions under J without the last element
  int j;
  int sz=size(rez); // their number
  list P;
  int last;
  for(i=1; i<=sz; i++) // for every such partition
  {
    P = rez[i];
    last = P[size(P)];
    for(j = last;j<=Jlast ;j++) // for every number between its last entry and the last entry of J
    {
      // append that number to the end of the partition
      // and append the resulting partition to the final result
      rez1 = rez1 + list(P+list(j));
    }
  }
  return(rez1);
}
example
{
  "EXAMPLE:"; echo =2;
  // Partitions under the partition (0, 1, 1)
  list I = 0, 1, 1;
  print( partUnder(I) );
}
//----------------------------------------------------------------------------------------
// The procedures below are for the internal usage only
//----------------------------------------------------------------------------------------

static proc append_by_zeroes(int N, list c)
"USAGE:   append_by_zeroes( N, c);  N integer, c a list
RETURN:   list
PURPOSE:  appends by zeroes up to the length N
EXAMPLE:  example append_by_zeroes; shows an example
NOTE:
"
{
  int n=size(c);
  if(N>n) // if N is greater than the length of c, append c by zeroes up to the length N
  {
    int i;
    for(i=n+1;i<=N;i++)
    {
      c=c+list( poly(0) );
    }
  }
  return(c);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (x, y, z), dp;
  list l=(x, y, z);
  //append the list by two zeroes and get a list of lenght 5
  print( append_by_zeroes(5, l) );
}
//-----------------------------------------------------------------------

static proc is_integer(def r)
"USAGE:   is_integer(r);  r any type
RETURN:   1 or 0
PURPOSE:  checks whether r is of type int or bigint
EXAMPLE:  example is_integer; shows an example
NOTE: returns 1 if r is of type int or bigint, otherwise returns 0
"
{
  if( (typeof(r)=="int") || (typeof(r)=="bigint") )
  {
    return(1);
  }
  else
  {
    return(0);
  }
}
example
{
  "EXAMPLE:";echo =2;
  // test on int, bigint, poly
  ring r;
  int i=12;
  bigint j=16;
  poly f=x;
  print( is_integer(i) );
  print( is_integer(j) );
  print( is_integer(f) );
}
//------------------------------------------------------------------------------------

static proc integer_list(list l)
"USAGE:   integer_list(l);  l list
RETURN:   list
PURPOSE:  gets the first positive ingerer entries of l, computes their maximum;
          used for adjusting the lists of optional parameters that are suposed to be integers
EXAMPLE:  example integer_list; shows an example
NOTE:     used in chWedge(...) and chSymm(...)
"
{
  int M=0;
  int n=size(l);
  if(n==0)
  {
    return(list(l, M));
  }
  // now n>0
  list rez; // the result will be computed here
  int i=1;
  while( is_integer( l[i] ) ) // take only the first integer entries of l
  {
    if(l[i]>0) // if they are positive
    {
      rez=rez+list( l[i] );
      if(l[i]>M) // adjust the maximum if necessary
      {
        M=l[i];
      }
      i++;
    }
    else // otherwise get out from the loop
    {
      break;
    }
  }
  return( list( rez, M)   );
}
example
{
  "EXAMPLE:";echo =2;
  // the first integer entries of 1,2,3,t are 1,2,3
  ring r=0,(t), ls;
  list l=1,2,3, t;
  print( integer_list(l) );
}
//---------------------------------------------------------------------------------------

static proc appendToAll(list L, list A)
"USAGE:   appendToAll( L, A );  L list of lists, A list
RETURN:   list
PURPOSE:  appends A to every entry of L
EXAMPLE:  example appendToAll; shows an example
NOTE:
"
{
  int n=size(L);
  int i;
  for(i=1;i<=n;i++) // run through all elements of L
  {
    L[i]=L[i]+A; // and append A to each of them
  }
  return(L);
}
example
{
  "EXAMPLE:"; echo=2;
  // Consider two lists
  list l1, l2;
  l1=1,2;
  l2=3,4;
  // The first one is
  print(l1);
  // The second one is
  print(l2);
  // Now consider the list with entries l1 and l2
  list L= l1, l2;
  print(L);
  // and consider a list A
  list A = 7,9;
  print(A);
  // append A to all entries of L
  print( appendToAll(L, A) );
}
//---------------------------------------------------------------------------------------

static proc listSame(int n, int k)
"USAGE:   listSame( n, k );  n integer, k non-negative integer
RETURN:   list
PURPOSE:  list with k entries each equal to n
EXAMPLE:  example listSame; shows an example
NOTE:     if k is negative or zero, the empty list is returned
"
{
  list rez;
  int i;
  for(i=1;i<=k;i++) // create a list with k entries, each equal to n
  {
    rez=rez+list(n);
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo=2;
  // list of 5 zeroes
  print( listSame(0, 5) );
}
//---------------------------------------------------------------------------------------

static proc IJcoef(list I, list J)
"USAGE:   IJcoef( I, J);  J, J lists of integers
RETURN:   bigint
PURPOSE:  computes the coefficient used in the formula of Lascoux
EXAMPLE:  example IJcoef; shows an example
NOTE:     these coefficients are denoted (I, J) in the paper of Lascoux
"
{
  int m = size(I);
  if(m != size(J)) // if the sizes of I and J are different
  {
    // give a warning
    print("The sizes of the partitions are different!");
    print("Zero is returned.");
    return( bigint(0) ); // and return zero
  }
  // now the sizes of I and J are equal m
  int h, k;
  bigintmat M[m][m]; // construct the required matrix
  for(h=1; h<=m; h++)
  {
    for(k=1; k<=m; k++)
    {
      M[h,k] = binomial( I[k]+k-1, J[h]+h-1 );
    }
  }
  return( det(M) ); // and return its determinant
}
example
{
  "EXAMPLE:"; echo =2;
  // The coefficient corresponding to the partitions (1, 3, 4) and (0, 3, 3)
  list I = 1, 3, 4;
  list J = 1, 3, 3;
  print( IJcoef(I, J) );
}
//---------------------------------------------------------------------------------------

static proc invertPart(list l)
"USAGE:   invertPart(I);  I list of integers (partition),
RETURN:   list of integers
PURPOSE:  inverts the ordering of the elements in the list
EXAMPLE:  example invertPart; shows an example
NOTE:
"
{
  list L;
  int sz=size(l);
  int i;
  for(i=sz;i>=1;i--)
  {
    L=L+list(l[i]);
  }
  return(L);
}
example
{
  "EXAMPLE:"; echo = 2;
  // Invert the ordering of elements in (3, 2, 1)
  list l = 3, 2, 1;
  print( invertPart(l) );
}
//---------------------------------------------------------------------------------------

static proc LRmul(list I, list J)
"USAGE:   LRmul(x, y);  x, y lists of integers (partitions)
RETURN:   list of lists
PURPOSE:  computes the partitions z for which the Littlewood-Richardson
          coefficient c^z_{x,y} is non-zero together with that coefficient;
          partitions up to length r
EXAMPLE:  example LRmul; shows an example
NOTE:     uses LRmult(..) from lrcalc.lib, does the same,
          only uses the inverted ordering of the elements in the partition
"
{
  list rez=LRmult(invertPart(I), invertPart(J));
  int sz=size(rez);
  int i;
  for(i=1;i<=sz;i++)
  {
    rez[i][2]=invertPart(rez[i][2]);
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo = 2;
  // Compute the partitions z for which the Littlewood-Richardson coefficient
  // c^z_{x,y} is non-zero together with that coefficient
  // for x= (1, 2), y=(1, 2)
  list x = 1, 2;
  list y = 1, 2;
  print( LRmul(x, y) );
}
//---------------------------------------------------------------------------------------

static proc hook(list I)
"USAGE:   hook(I);  I list of integers (partition),
RETURN:   bigint
PURPOSE:  computes the product of the hook lenhths of the partition I
EXAMPLE:  example hook; shows an example
NOTE:
"
{
  bigint rez=1;
  list dI= invertPart( dualPart(I) );
  I=invertPart( I );
  int szI=size(I);
  int szdI=size(dI);
  int i, j;
  for(i=1;i<=szI;i++)
  {
    for(j=1;j<=I[i];j++)
    {
      rez=rez*(I[i]+dI[j]-i-j+1);
    }
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo = 2;
  // compute the product of all hook lengths of the partition (1, 1, 3)
  list I = 1, 1, 3;
  print( hook(I) );
}
//---------------------------------------------------------------------------------------

static proc apn0_int(int N, list c)
"USAGE:   apn0_int( N, c);  N integer, c list of integers (partition)
RETURN:   list of integers
PURPOSE:  appends by integer zeroes up to the length N
EXAMPLE:  example apn0_int; shows an example
NOTE:
"
{
  int n=size(c);
  if(N>n) // if N is greater than the length of c, append c by zeroes up to the length N
  {
    int i;
    for(i=n+1;i<=N;i++)
    {
      c=c+list( int(0) );
    }
  }
  return(c);
}
example
{
  "EXAMPLE:";echo =2;
  ring r = 0, (x, y, z), dp;
  list l=(1, 2, 3);
  //append the list by two zeroes and get a list of lenght 5
  print( apn0_int(5, l) );
}
//---------------------------------------------------------------------------------------

static proc contentPoly(list I, list J, poly e)
"USAGE:   contentPoly(I, J, e);  L, M lists of integers (partitions),
          e polynomial
RETURN:   poly
PURPOSE:  computes the content polynomial of the skew partition corresponding to I>J
EXAMPLE:  example contentPoly; shows an example
NOTE:
"
{
  int i, j;
  int szI=size(I);
  I=invertPart(I);
  J=invertPart(J);
  J=apn0_int(szI, J);
  poly rez=1;
  for(i=1; i<=szI; i++)
  {
    for(j=J[i]+1; j<=I[i]; j++)
    {
      rez = rez*(e- i+j);
    }
  }
  return(rez);
}
example
{
  "EXAMPLE:"; echo = 2;
  // compute the content Polynomial of the skew partition
  // corresponding to  (1,2) > (0, 1) with respect to the variable x
  ring r = 0, (x), dp;
  list L=1,2;
  list M=0,1;
  print( contentPoly(L, M, x) );
}
//---------------------------------------------------------------------------------------

static proc Pcoef(list L, list M, poly e, poly f)
"USAGE:   Pcoef(L, M, e, f);  L, M lists of integers (partitions),
          e, f polynomials
RETURN:   poly
PURPOSE:  computes the polynomial P_{L, M}(e, f) from the paper of Manivel
EXAMPLE:  example Pcoef; shows an example
NOTE:
"
{
  list P = LRmul(dualPart(L), M);
  int  sz=size(P);
  poly rez;
  //poly h;
  list T, DT;
  //bigint lrc;
  int i;
  for(i=1;i<=sz;i++)
  {
    T=P[i][2];
    DT=dualPart(T);
    //lrc=P[i][1];
    //h=contentPolyM(DT, L, e)*contentPoly(T, M, f);
    rez=rez+P[i][1]* contentPoly(DT, L, e)*contentPoly(T, M, f)/hook(DT);
  }
 return(rez);
}
example
{
  "EXAMPLE:"; echo = 2;
  // compute P_{L, M}(e, f) from the paper of Manivel
  // for L = (0,1) and M = (1, 1)
  ring r = 0, (e, f), dp;
  list L=1,2,3;
  list M=1,2;
  print( Pcoef(L, M, e, f) );
}
//---------------------------------------------------------------------------------------