source: git/Singular/LIB/hyperel.lib @ 7d56875

///////////////////////////////////////////////////////////////////////////////
version="$Id: hyperel.lib,v 1.1 2007-05-24 11:57:58 Singular Exp $";
category="Teaching";
info="
LIBRARY:    hyperelliptic.lib
AUTHOR:     Markus Hochstetter, markushochstetter@gmx.de

 NOTE: This library provides procedures for computing with divisors in the
       jacobian of hyperelliptic curves. In addition procedures are available
       for computing the rational representation of divisors and vice versa.
       The library is intended to be used for teaching and demonstrating
       purposes but not for efficient computations.




PROCEDURES:
  ishyper(h,f)           test, if y^2+h(x)y=f(x) is hyperelliptic
  isoncurve(P,h,f)       test, if point P is on C: y^2+h(x)y=f(x)
  chinrestp(b,moduli)    compute polynom x, s.t. x=b[i] mod moduli[i]
  norm(a,b,h,f)          norm of a(x)-b(x)y in IF[C]
  multi(a,b,c,d,h,f)     (a(x)-b(x)y)*(c(x)-d(x)y)  in IF[C]
  ratrep (P,h,f)         returns polynomials a,b, s.t. div(a,b)=P
  divisor(a,b,h,f,[])    computes divisor of a(x)-b(x)y
  gcddivisor(p,q)        gcd of the divisors p and q
  semidiv(D,h,f)         semireduced divisor of the pair of polys D[1], D[2]
  cantoradd(D,Q,h,f)     adding divisors of the hyperell. curve y^2+h(x)y=f(x)
  cantorred(D,h,f)       returns reduced divisor which is equivalent to D
  double(D,h,f)          computes 2*D on y^2+h(x)y=f(x)
  cantormult(m,D,h,f)    computes m*D on y^2+h(x)y=f(x)

              [parameters in square brackets are optional]
";
///////////////////////////////////////////////////////////////////////////////


//=============== Test, if a given curve is hyperelliptic =====================

proc ishyper(poly h, poly f)
"USAGE:   ishyper(h,f); h,f poly
RETURN:  1 if y^2+h(x)y=f(x) is hyperelliptic, else 0
NOTE:    Tests, if y^2+h(x)y=f(x) is a hyperelliptic curve.
         Curve is defined over basering. Additionally shows error-messages.
EXAMPLE: example ishyper; shows an example
"
{
  // constructing a copy of the basering (only variable x),
  // with variables x,y.
  def R=basering;
  list l= ringlist(R);
  list ll=l[2];
  ll="x","y";
  l[2]=ll;
  intvec v= l[3][1][2];
  v=v,1;
  l[3][1][2]=v;
  def s=ring(l);
  setring s;

  // test, if y^2 + hy - f is hyperelliptic.
  int i=1;
  poly h=imap(R,h);
  poly f=imap(R,f);
  poly F=y2 + h*y - f;
  ideal I=F, diff(F,x) , diff(F,y);
  ideal J=std(I);
  if ( J != 1 )
  {
      i=0;
      "The curve is singular!";
  }
  if ( deg(f) mod 2 != 1 )
  {
     i=0;
      "The polynomial ",f," has even degree!";
  }
  if ( leadcoef(f) != 1 )
  {
     i=0;
     "The polynomial ",f," is not monic!";
  }
  if ( 2*deg(h) > deg(f)-1 )
  {
    i=0;
     "The polynomial ",h," has degree ",deg(h),"!";
  }
  setring(R);
  return(i);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   ishyper(h,f);
}


//================= Test, if a given ponit is on the curve ====================

proc isoncurve(list P, poly h, poly f)
"USAGE:   isoncurve(P,h,f); h,f poly
RETURN:  1 or 0 (if P is on curve or not)
NOTE:    Tests, if P=(P[1],P[2]) is on the hyperelliptic curve
         y^2+h(x)y=f(x) .
         Curve is defined over basering.
EXAMPLE: example isoncurve; shows an example
"
{
   if ( P[2]^2 + subst(h,var(1),P[1])*P[2] - subst(f,var(1),P[1]) == 0 )
   {
      return(1);
   }
   return(0);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   list P=2,3;
   isoncurve(P,h,f);
}


//====================== Remainder of a polynomial division ===================

proc divrem(f,g)
"USAGE:   divrem(f,g);  f,g poly
RETURN:  remainder of the division f/g
NOTE:    Computes R, s.t. f=a*g + R, and deg(R) < deg(g)
EXAMPLE: example divrem; shows an example
"
{
   return(reduce(f,std(g)));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=0,x,dp;
   divrem(x2+1,x2);
}


//================ chinese remainder theorem for polynomials ==================

proc chinrestp(list b,list moduli)
"USAGE:   chinrestp(b,moduli);  b, moduli list of polynomials
RETURN:  polynomial x, s.t. x= b[i] mod moduli[i]
NOTE:    chinese remainder theorem for polynomials
EXAMPLE: example chinrestp; shows an example
"
{
   int i;
   int n=size(moduli);
   poly M=1;
   for(i=1;i<=n;i++)
   {
      M=M*moduli[i];
   }
   list m;
   for(i=1;i<=n;i++)
   {
      m[i]=M/moduli[i];
   }
   list y;
   for(i=1;i<=n;i++)
   {
      y[i]= extgcd(moduli[i],m[i])[3];
   }
   poly B=0;
   for(i=1;i<=n;i++)
   {
      B=B+y[i]*m[i]*b[i];
   }
   B=divrem(B,M);
   return(B);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   list b=3x-4, -3x2+1, 1, 4;
   list moduli=(x-2)^2, (x-5)^3, x-1, x-6;
   chinrestp(b,moduli);
}


//========================= norm of a polynomial ===============================

proc norm(poly a, poly b, poly h, poly f)
"USAGE:   norm(a,b,h,f);
RETURN:  norm of a(x)-b(x)y in IF[C]
NOTE:    The norm is a polynomial in just one variable.
         Curve C:y^2+h(x)y=f(x) is defined over basering.
EXAMPLE: example norm; shows an example
"
{
   poly n=a^2+a*b*h-b^2*f;
   return(n);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   poly a=x2+1;
   poly b=x;
   norm(a,b,h,f);
}


//========== multiplikation of polynomials in the coordinate ring =============

proc multi(poly a, poly b, poly c, poly d, poly h, poly f)
"USAGE:   multi(a,b,c,d,h,f);
RETURN:  list L with L[1]-L[2]y=(a(x)-b(x)y)*(c(x)-d(x)y) in IF[C]
NOTE:    Curve C:y^2+h(x)y=f(x) is defined over basering.
EXAMPLE: example multi; shows an example
"
{
   poly A=a*c + b*d*f;
   poly B=b*c +a*d + b*h*d;
   return (list(A,B));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // hyperelliptic curve y^2 + h*y = f
   poly a=x2+1;
   poly b=x;
   poly c=5;
   poly d=-x;
   multi(a,b,c,d,h,f);
}


//================== polynomial expansion around a point ========================

proc darst(list P,int k, poly h, poly f)
"USAGE:   darst(P,k,h,f);
RETURN:  list c of length k
NOTE:    expansion around point P in IF[C], s.t.
         y=c[1]+c[2]*(x-P[1]) +...+c[k]*(x-P[1])^k-1 + rest.
         Curve C:y^2+h(x)y=f(x) is defined over basering.
EXAMPLE: example darst; shows an example
"
{

   if ( P[2] == -P[2]- subst(h,var(1),P[1]))
   {
    ERROR("no special points allowed");
   }
   list c;
   list r;
   list n;
   poly N;
   c[1]=P[2];
   r[1]=list(0,-1,1,0);
   poly r1,r2,r3,r4;
   // rational function are represented as (r1 - r2*y) / (r3 - r4*y)

   for (int i=1; i<k ; i++)
   {
      r1=r[i][1]-c[i]*r[i][3];
      r2=r[i][2]-c[i]*r[i][4];
      r3=r[i][3];
      r4=r[i][4];
      n=multi(r3,r4,r1+r2*h,-r2,h,f);
      N=r1*r1 + r1*r2*h-r2*r2*f;
      r[i+1]=list(N/(var(1)-P[1]),0,n[1],n[2]);
      while ((divrem(r[i+1][1],var(1)-P[1]) ==0) and (divrem(r[i+1][2],var(1)-P[1]) ==0) and (divrem(r[i+1][3],var(1)-P[1]) ==0) and (divrem(r[i+1][4],var(1)-P[1]) ==0))
      {
          // reducing the rationl function
          //(r[i+1][1] - r[i+1][2]*y)/(r[i+1][3] - r[i+1][4]) , otherwise there
          // could be a pole, s.t. conditions are not fulfilled.
          r[i+1][1]=(r[i+1][1]) / (var(1)-P[1]);
          r[i+1][2]=(r[i+1][2]) / (var(1)-P[1]);
          r[i+1][3]=(r[i+1][3]) / (var(1)-P[1]);
          r[i+1][4]=(r[i+1][4]) / (var(1)-P[1]);
      }
      c[i+1]=(subst(r[i+1][1],var(1),P[1]) - subst(r[i+1][2],var(1),P[1])*P[2]) / (subst(r[i+1][3],var(1),P[1]) - subst(r[i+1][4],var(1),P[1])*P[2]);
   }
   return(c);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   list P=5,3;
   darst(P,3,h,f);
}


//================ rational representation of a divisor =======================

proc ratrep1 (list P, poly h, poly f)
"USAGE:   ratrep1(P,k,h,f);
RETURN:  list (a,b)
NOTE:    Important: P has to be semireduced!
         Computes rational representation of the divisor
         P[1][3]*(P[1][1], P[1][2]) +...+ P[sizeof(P)][3]*
         *(P[sizeof(P)][1], P[sizeof(P)][2]) - (*)infty=div(a,b)
         Divisor P has to be semireduced.
         Curve C:y^2+h(x)y=f(x) is defined over basering.
SEE AlSO: ratrep
EXAMPLE: example ratrep1; shows an example
"
{
   poly a=1;
   list b;
   list m;
   list koef;
   int k;

   // Determination of the polynomial b[i] for each point using procedure darst
   for (int i=1 ; i<= size(P); i++)
   {
      a=a*(var(1)-P[i][1])^(P[i][3]);    // computing polynomial a
      m[i]=(var(1)-P[i][1])^(P[i][3]);
      b[i]=P[i][2];
      k=1;

      while (divrem(b[i]*b[i] + b[i] *h - f,(var(1)-P[i][1])^(P[i][3])) != 0)
      {
          k=k+1;   // b[i]=P[i][2];
          koef=darst(list (P[i][1],P[i][2]), k, h,f);
          // could be improved, if one doesn't compute list coef completely new
          // every time
          b[i]=b[i]+ koef[k]*(var(1)-P[i][1])^(k-1);
      }
   }
   // Return polynomial a and b. Polynomial b is solution of the congruencies
   // b[i] mod m[i] .
   return(list(a,chinrestp(b,m)));

}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   //divisor P
   list P=list(-1,-3,1),list(1,1,1);
   ratrep1(P,h,f);
}


//================ rational representation of a divisor =======================

proc ratrep (list P, poly h, poly f)
"USAGE:   ratrep(P,k,h,f);
RETURN:  list (a,b)
NOTE:    Importatnt: P has to be semireduced!
         Computes rational representation of the divisor
         P[1][3]*(P[1][1], P[1][2]) +...+ P[sizeof(P)][3]*
         *(P[sizeof(P)][1], P[sizeof(P)][2]) - (*)infty=div(a,b)
         Divisor P has to be semireduced.
         Curve C:y^2+h(x)y=f(x) is defined over basering.
         Works faster than ratrep1.
SEE ALSO: ratrep1
EXAMPLE: example ratrep; shows an example
"
{
   poly a=1;
   list b;
   list m;
   list koef;
   int k;

   poly c;
   list r;
   list n;
   poly Norm;
   poly r1,r2,r3,r4;

   // Determination of the polynomial b[i] for each point using procedure darst
   for (int i=1 ; i<= size(P); i++)
   {
      a=a*(var(1)-P[i][1])^(P[i][3]);     // computing polynomial a
      m[i]=(var(1)-P[i][1])^(P[i][3]);
      b[i]=P[i][2];
      k=1;
      c=P[i][2];
      r=0,-1,1,0;
      while (divrem(b[i]*b[i] + b[i] *h - f,(var(1)-P[i][1])^(P[i][3])) != 0)
      {
          k=k+1;
          // here, the procedure darst was integrateg. In every pass a new
          // coefficient c[i] is determined.
          r1=r[1]-c*r[3];
          r2=r[2]-c*r[4];
          r3=r[3];
          r4=r[4];
          n=multi(r3,r4,r1+r2*h,-r2,h,f);
          Norm=r1*r1 + r1*r2*h-r2*r2*f;
          r=list(Norm/(var(1)-P[i][1]),0,n[1],n[2]);
          while ((divrem(r[1],var(1)-P[i][1]) ==0) and (divrem(r[2],var(1)-P[i][1]) ==0) and (divrem(r[3],var(1)-P[i][1]) ==0) and (divrem(r[4],var(1)-P[i][1]) ==0))
          {

           // reducing the rationl function
           // (r[1]-r[2]y)/(r[3]-r[4]y) , otherwise there
           // could be a pole, s.t. conditions are not fulfilled.
              r[1]=(r[1]) / (var(1)-P[i][1]);
              r[2]=(r[2]) / (var(1)-P[i][1]);
              r[3]=(r[3]) / (var(1)-P[i][1]);
              r[4]=(r[4]) / (var(1)-P[i][1]);
          }
          c=(subst(r[1],var(1),P[i][1]) - subst(r[2],var(1),P[i][1])*P[i][2]) / (subst(r[3],var(1),P[i][1]) - subst(r[4],var(1),P[i][1])*P[i][2]);
          b[i]=b[i]+ c*(var(1)-P[i][1])^(k-1);
      }
   }
   // return polynomial a and b. Polynomial b is solution of the congruencies
   // b[i] mod m[i] .
   return(list(a,chinrestp(b,m)));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   //Divisor P
   list P=list(-1,-3,1),list(1,1,1);
   ratrep(P,h,f);
}


//============== Order of a zero in a polynomial ==============================

proc ordnung(poly x0 , poly g)
"USAGE:   ordnung(x0,g);
RETURN:  int i
NOTE:    i is maximal, s.t. (x-x0)^i divides g
EXAMPLE: example ordnung; shows an example
"
{
   poly gg=g;
   int i;
   while ( divrem(gg,var(1)-x0) ==0 )
   {
      i=i+1;
      gg=gg/(var(1)-x0);
   }
   return(i);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=0,x,dp;
   poly g=(x-5)^7*(x-3)^2;
   number x0=5;
   ordnung(x0,g);
}


//================== divisor of a polynomial function =========================

proc divisor(poly a, poly b, poly h, poly f, list #)
"USAGE:   divisor(a,b,h,f); optional: divisor(a,b,h,f,s); s=0,1
RETURN:  list P
NOTE:    P[1][3]*(P[1][1], P[1][2]) +...+ P[sizeof(P)][3]*
         *(P[sizeof(P)][1], P[sizeof(P)][2]) - (*)infty=div(a(x)-b(x)y)
         if there is an optional parameter s!=0, then divisor additonally
         returns a parameter, which says, whether irreducible polynomials
         occured during computations or not. Otherwise only warnings are
         displayed on the monitor. For s=0 nothing happens.
         Curve C:y^2+h(x)y=f(x) is defined over basering.
EXAMPLE: example divisor; shows an example
"
{
   list p;
   int j;
   poly x0;
   list y;
   list fa=factorize(gcd(a,b));  // wanted: common roots of a and b
   poly Norm=norm(a,b,h,f);

   int s;
   int irred=0;
   if (size(#)>0)
   {
      s=#[1];
   }
   else
   {
      s=0;
   }

   for (int i=2; i<=size(fa[1]) ; i++)
   {
      // searching roots by finding polynomials of degree 1
      if ( deg(fa[1][i]) !=1 )
      {
         if (s==0)
         {
         "WARNIG: ", fa[1][i], "is irreducible over this field !";
         }
         else
         {
            irred=1;
         }
      }
      else
      {
         x0=var(1) - fa[1][i];
         // finding the y-coordinates; max. 2
         y= factorize(var(1)^2 + var(1)*subst(h,var(1),x0) - subst(f,var(1),x0));
         if ( deg(y[1][2]) == 1)
         // if root belongs to point on curve, then...
         {
            // compute order of a-b*y in the founded point
            j=j+1;
            p[j]=list(x0,var(1)-y[1][2],fa[2][i]);
            if ( y[2][2]== 1)          // ordinary point
            {
               j=j+1;
               p[j]=list(x0 , var(1)-y[1][3] , fa[2][i] );
               if (a/(var(1)-x0)^(fa[2][i]) - b/(var(1)-x0)^(fa[2][i]) * p[j][2] ==0 )
               {
                  p[j][3]= p[j][3] + ordnung(x0,norm(a/(var(1)-x0)^(fa[2][i]) , b/(var(1)-x0)^(fa[2][i]),h,f));
               }
               if (a/(var(1)-x0)^(fa[2][i]) - b/(var(1)-x0)^(fa[2][i]) * p[j-1][2] ==0 )
               {
               p[j-1][3]=p[j-1][3] + ordnung(x0,norm(a/(var(1)-x0)^(fa[2][i]) , b/(var(1)-x0)^(fa[2][i]),h,f));
               }
            }
            else        // special point
            {
               p[j][3]=p[j][3] *2  ;
               if (a/(var(1)-x0)^(fa[2][i]) - b/(var(1)-x0)^(fa[2][i]) * p[j][2] ==0 )
               {
                  p[j][3]=p[j][3] + ordnung(x0,norm(a/(var(1)-x0)^(fa[2][i]) , b/(var(1)-x0)^(fa[2][i]),h,f));
               }
            }

         }
         // Norm of a-b*y is reduced by common root of a and b
         // (is worked off)
         Norm = Norm/((var(1)-x0)^(ordnung(x0,Norm)));
      }
   }

   // some points are still missing; points for which a and b have no common
   // roots, but norm(a-b*Y)=0 .
   fa=factorize(Norm);
   for ( i=2 ; i<=size(fa[1]) ; i++)
   {
      if ( deg(fa[1][i]) !=1)
      {
         if (s==0)
         {
         "WARNING: ", fa[1][i], "is irreducible over this field !";
         }
         else
         {
            irred=1;
         }
      }
      else
      {
         x0=var(1) - fa[1][i];
         y= factorize(var(1)^2 + var(1)*subst(h,var(1),x0) - subst(f,var(1),x0));
         if ( deg(y[1][2]) == 1)
         // if root belongs to point on curve, then...
         {
            if (subst(a,var(1),x0)- subst(b,var(1),x0)* (var(1)-y[1][2]) ==0)
            {
               p[size(p)+1]=list(x0,var(1)-y[1][2], ordnung(x0,Norm,h,f));
            }
            if ( y[2][2]== 1)     // ordinary point
            {
               if (subst(a,var(1),x0)- subst(b,var(1),x0)* (var(1)-y[1][3]) ==0)
               {
               p[size(p)+1]=list(x0 , var(1)-y[1][3] , ordnung(x0,Norm,h,f));
               }
            }
         }
      }
   }
   if (s==0)
   {
      return(p);
   }
   return(p,irred);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   poly a=(x-1)^2*(x-6);
   poly b=0;
   divisor(a,b,h,f,1);
}


//===================== gcd of two divisors ===================================

proc gcddivisor(list p, list q)
"USAGE:   gcddivisor(p,q);
RETURN:  list P
NOTE:    gcd of two divisors
EXAMPLE: example gcddivisor; shows an example
"
{
   list e;
   int i,j;
   for (i=1 ; i<= size(p) ; i++)
   {
      for (j=1 ; j<= size(q) ; j++)
      {
         if ( p[i][1] == q[j][1] and p[i][2] == q[j][2])
         {
            if ( p[i][3] <= q[j][3] )
            {
               e[size(e)+1]= list (p[i][1] , p[i][2] , p[i][3]);
            }
            else
            {
               e[size(e)+1]= list (q[j][1] , q[j][2] , q[j][3]);
            }
         }
      }
   }
   return(e);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // two divisors
   list p=list(-1,-3,1),list(1,1,2);
   list q=list(1,1,1),list(2,2,1);
   gcddivisor(p,q);
}


//========== semireduced divisor from rational representation=================

proc semidiv(list D,poly h, poly f)
"USAGE:   semidiv(D,h,f);
RETURN:  list P
NOTE:    important: Divisor D has to be semireduced!
         Computes semireduced divisor
         P[1][3]*(P[1][1], P[1][2]) +...+ P[sizeof(P)][3]*
         *(P[sizeof(P)][1], P[sizeof(P)][2]) - (*)infty=div(D[1],D[2])
         Curve C:y^2+h(x)y=f(x) is defined over basering.
EXAMPLE: example semidiv; shows an example
"
{
   if ( deg(D[2]) >= deg(D[1]) or divrem(D[2]^2+D[2]*h-f,D[1]) != 0 )
   {
      ERROR("Pair of polynomials doesn't belong to semireduced divisor!");
   }
   list D1,D2;
   int s1,s2;
   D1,s1=divisor(D[1],0,h,f,1);
   D2,s2=divisor(D[2],1,h,f,1);

   // Only if irreducible polynomials occured in D1 !and! D2 a warning
   // is necessary.
   if (s1==1 and s2==1)
   {
      "Attention:
     Perhaps some points were not found over this field!";
   }
   return(gcddivisor(D1,D2));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // Divisor
   list D=x2-1,2x-1;
   semidiv(D,h,f);
}


//=============== Cantor's algorithm - composition ============================

proc cantoradd(list D, list Q, poly h, poly f)
"USAGE:   cantoradd(D,Q,h,f);
RETURN:  list P
NOTE:    Cantor's Algorithm - composition
         important: D and Q have to be semireduced!
         Computes semireduced divisor
         div(P[1],P[2])= div(D[1],D[2]) + div(Q[1],Q[2])
         The divisors are defined over the basering.
         Curve C:y^2+h(x)y=f(x) is defined over the basering.
EXAMPLE: example cantoradd; shows an example
"
{
   poly a;
   poly b;
   list e=extgcd(D[1],Q[1]);
   if ( e[1]==1 )
   {
      a=D[1]*Q[1];
      b=divrem( e[2]*D[1]*Q[2] + e[3]*Q[1]*D[2] ,a);
      return(list(a,b));
   }
   list c=extgcd(e[1],D[2]+Q[2]+h);
   poly s1=e[2]*c[2];
   poly s2=c[2]*e[3];
   poly s3=c[3];
   a=D[1]*Q[1]/c[1]^2;
   b=divrem((s1*D[1]*Q[2] + s2*Q[1]*D[2] + s3*(D[2]*Q[2] + f))/c[1],a);
   return(list(a,b));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // two divisors in rational representation
   list D=x2-1,2x-1;
   list Q=x2-3x+2,-3x+1;
   cantoradd(D,Q,h,f);
}


//==================== Cantor's algorithm - reduction =========================

proc cantorred(list D,poly h,poly f)
"USAGE:   cantorred(D,h,f);
RETURN:  list N
NOTE:    Cantor's algorithm - reduction.
         important: Divisor D has to be semireduced!
         Computes reduced divisor div(N[1],N[2])= div(D[1],D[2]).
         The divisors are defined over the basering.
         Curve C:y^2+h(x)y=f(x) is defined over the basering.
EXAMPLE: example cantorred; shows an example
"
{
   list N=D;
   while ( 2*deg(N[1]) > deg(f)-1 )
   {
      N[1]=(f - N[2]*h - N[2]^2)/N[1];
      N[2]=divrem(-h-N[2],N[1]);
   }
   N[1]=N[1]/leadcoef(N[1]);
   return(N);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // semireduced divisor
   list D=2x4+3x3-3x-2, -x3-2x2+3x+1;
   cantorred(D,h,f);
}


//================= doubling a semireduced divisor ============================

proc double(list D, poly h, poly f)
"USAGE:   double(D,h,f);
RETURN:  list Q=2*D
NOTE:    important: Divisor D has to be semireduced!
         Special case of Cantor's algorithm.
         Computes reduced divisor div(Q[1],Q[2])= 2*div(D[1],D[2])
         The divisors are defined over the basering.
         Curve C:y^2+h(x)y=f(x) is defined over the basering.
EXAMPLE: example double; shows an example
"
{
   list c=extgcd(D[1], 2*D[2] + h);
   poly a=D[1]*D[1]/c[1]^2;
   poly b=divrem((c[2]*D[1]*D[2] + c[3]*(D[2]*D[2] + f))/c[1],a);
   return(cantorred(list(a,b),h,f));
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // reduced divisor
   list D=x2-1,2x-1;
   double(D,h,f);
}


//================ multiples of a semireduced divisor =========================

proc cantormult(int m, list D, poly h, poly f)
"USAGE:   cantormult(m,D,h,f);
RETURN:  list res=m*D
NOTE:    important: Divisor D has to be semireduced!
         Uses repeated doublings for a faster computation
         of the reduced divisor m*D.
         Attention: Factor m is int, this means bounded.
         For m<0 the inverse of m*D is returned.
         The divisors are defined over the basering.
         Curve C:y^2+h(x)y=f(x) is defined over the basering.
EXAMPLE: example cantormult; shows an example
"
{
   list res=1,0;
   list bas=D;
   int exp=m;
   if (exp==0) { return(list(1,0)); }
   if (exp==1) { return(D); }
   if (exp==-1) { return(list(D[1],-D[2]-h)) ; }
   if ( exp < 0)
   {
      exp=-exp;
   }
   while ( exp > 0 )
   {
     if ( (exp mod 2) !=0 )
     {
       res = cantorred(cantoradd(res,bas,h,f),h,f);
       exp=exp-1;
     }
     bas=double(bas,h,f);
     exp=exp/2;
   }
   if ( m < 0 )
   {
      res[2]=-res[2]-h;
   }
   return(res);
}
example
{ "EXAMPLE:"; echo = 2;
   ring R=7,x,dp;
   // hyperelliptic curve y^2 + h*y = f
   poly h=x;
   poly f=x5+5x4+6x2+x+3;
   // reduced divisor
   list D=x2-1,2x-1;
   cantormult(34,D,h,f);
}


/*
//=============================================================================
//   In the following you find a large example, which demostrates the use of
//   the most important procedures.
//=============================================================================
//---- field with 2^5=32 elements ----
ring r=(2,a),x,dp;
minpoly=a5+a2+1;

//---- hyperelliptic curve y^2 + hy = f ----
poly h=x2+x;
poly f=x5+x3+1;

//---- two divisors ----
list l1=list(a30,0,1),list(0,1,1);
list l2=list(a30,a16,1),list(1,1,1);

//---- their rational representation ----
list D1=ratrep(l1,h,f); D1;
//[1]:
//   x2+(a4+a)*x
//[2]:
//   (a)*x+1

list D2=ratrep(l2,h,f); D2;
//[1]:
//   x2+(a4+a+1)*x+(a4+a)
//[2]:
//   (a3+a2+a+1)*x+(a3+a2+a)

//---- back to the point-based-representation ----
semidiv(D1,h,f);
//[1]:
//   [1]:
//      (a4+a)
//   [2]:
//      0
//   [3]:
//      1
//[2]:
//   [1]:
//      0
//   [2]:
//      1
//   [3]:
//      1

semidiv(D2,h,f);
//[1]:
//   [1]:
//      (a4+a)
//   [2]:
//      (a4+a3+a+1)
//   [3]:
//      1
//[2]:
//   [1]:
//      1
//   [2]:
//      1
//   [3]:
//      1

//---- adding D1 and D2 ----
list D12=cantorred(cantoradd(D1,D2,h,f),h,f);  D12;
//[1]:
//   x2+x
//[2]:
//   1

//---- D1+D2 in point-based-representation ----
semidiv(D12,h,f);
//[1]:
//   [1]:
//      1
//   [2]:
//      1
//   [3]:
//      1
//[2]:
//   [1]:
//      0
//   [2]:
//      1
//   [3]:
//      1

//---- D1 + D1   (2 possible ways, same result) ----
cantorred(cantoradd(D1,D1,h,f),h,f);
double(D1,h,f);
//[1]:
//   x2+(a3+1)
//[2]:
//   (a4+a3+a+1)*x+(a4+a3+a2+a+1)

//---- order of D1 in the jacobian over the basering ----
int i=1;
list E=D1;
while (E[1] != 1 or E[2] != 0 )
{
   E= cantorred(cantoradd(E,D1,h,f),h,f);
   i=i+1;
}
i;   // 482

//---- proof with multiplikation validates the result ----
cantormult(i,D1,h,f);
//[1]:
//   1
//[2]:
//   0

//---- computing the inverse of D1 ----
list d1= cantormult(-1,D1,h,f);  d1;
//[1]:
//   x2+(a4+a)*x
//[2]:
//   x2+(a+1)*x+1

//---- proof validates the result ----
cantoradd(d1,D1,h,f);
//[1]:
//   1
//[2]:
//   0


*/