# Package implementing NX-book, Chapter 5.4

 fgn := function(F) return [AlffGenus(F), AlffPlacesDegOneNum(F,1)]; end;
# fgn := function(F) return 0; end;

KEinit:=function(p,r)
 q := p^r;
 AlffInit(FF(p,2*r));
 K := [ Alff(y+T), Alff((y^q+y)*(T^(q-1)+1)+T^q) ];
 E := [ Alff(y+T), Alff(y^q+y+T^(q+1))           ]; 
end;

eest := function(n) return [ 
 Floor(q^n+q^(n-1)-q^((n+1)/2)-2*q^((n-1)/2)+1), (q^2-1)*q^(n-1)+1 ]; end;

Ktower:=function(itop)
 i := 2;
 Print("K[",1,"] : ",fgn(K[1]),"\n");
 Print("K[",i,"] : ",fgn(K[i]),"\n");
 while i<itop do 
  x     := AlffEltGenY( K[i] );
  p     := AlffEltMinPoly( x^q/(x^(q-1)+1) );
  i    := i+1;
  K[i] := Alff( Eval( p, y^q+y ) );
 Print("K[",i,"] : ",(K[i]),"\n");
 od;
end;

K2tower:=function(itop)
 i := 2;
 Print("K[",1,"] : ",fgn(K[1]),"\n");
 Print("K[",i,"] : ",fgn(K[i]),"\n");
 while i<itop do 
  x     := AlffEltGenY( K[i] );
  p     := find_minpol( x^q/(x^(q-1)+1) );
  i    := i+1;
  K[i] := Alff( Eval( p, y^q+y ) );
 Print("K[",i,"] : ",(K[i]),"\n");
 od;
end;


Etower:=function(itop)
 i:=2;
 Print("E[",1,"] : ",fgn(E[1]),"\n");
 Print("E*[",1,"] : ",eest(1),"\n");
 Print("E[",i,"] : ",fgn(E[i]),"\n");
 Print("E*[",i,"] : ",eest(i),"\n");
 while i<itop do 
  z     := AlffEltGenY( E[i] );
  p     := AlffEltMinPoly( z^q/(z^(q-1)+1) );
  i    := i+1;
  E[i] := Alff( Eval( p, y^q+y ) );
  Print("E[",i,"] : ",fgn(E[i]),"\n");
  Print("E*[",i,"] : ",eest(i),"\n");
 od;
end;

# e>1 !
Finit:=function(p,e)
 q := p^e;
 m := (q-1)/(p-1);
 AlffInit(FF(p,e));
 F := [ Alff(y+T) ];
end;

Ftower:=function(itop) 
 i:=1;
 Print("F[",i,"] : ",fgn(F[i]),"\n");
 while i<itop do 
  x    := AlffEltGenY( F[i] );
  p    := AlffEltMinPoly( 1-(x+1)^m );
  i    := i+1;
  F[i] := Alff( Eval( p, y^m ) );
  Print("F[",i,"] : ",fgn(F[i]),"\n");
 od; 
end;

#  p^e>2 !
F2init:=function(p,e)
 r := p^e;
 AlffInit(FF(p,2*e));
 F2 := [ Alff(y+T) ];
end;

F2tower:=function(itop) 
 i:=1;
 Print("F2[",i,"] : ",fgn(F2[i]),"\n");
 while i<itop do 
  x    := AlffEltGenY( F2[i] );
  p    := AlffEltMinPoly( 1-(x+1)^(r-1) );
  i    := i+1;
  F2[i] := Alff( Eval( p, y^(r-1) ) );
  Print("F2[",i,"] : ",fgn(F2[i]),"\n");
 od; 
end;


find_minpol := function(genK)
 local powlist,krnl,coeff;
# powlist:=List([0..deg-2],i->genK^i);
 powlist:=[genK^0];
 coeff:=[]; 
 i:=1;
 repeat
  Append(powlist,[powlist[i]*genK]);
  krnl:=MatKernel( Mat(kT, List(powlist,pow->AlffEltToList(pow)[1]) ) );
  if MatRows(krnl)>0 
   then coeff:=MatToRowList( krnl)[1]; 
  fi;
 until Length(coeff)>0; 
 powlist := List(powlist, e->AlffEltToList(e)[2]);
 powlist := Sum([0..Length(coeff)-1],i->coeff[i+1]*powlist[i+1]*y^i);
 return powlist/Gcd(PolyToList(powlist));
end;