//basic tower example///////////////////////////////////////////
//Given (x_1+1)x_1(x_2+1)=x_2^2 and (x_2+1)x_2(x_4+1)=x_4^2,char=2
//define x_6:=x_2(x_4+1) and x_7:=x_1(x_2+1)(x_4+1) to get a
//one-point curve description in terms of x_4,x_6,x_7,
//with subscripts corresponding to pole orders at P_{\infty}



F:=FiniteField(2);
WT_MATRIX_S:=[[7,4]];
S:=PolynomialRing(F,2,"weight",
[7,4,1,0]);

AssignNames(~S,
["x" cat &cat
[ "_" cat IntegerToString(WT_MATRIX_S[i,j]) 
: i in [1..#WT_MATRIX_S]]
: j in [1..#WT_MATRIX_S[1]]]);

WT_MATRIX_icS,integral_closure_S,I,delta,phi,psi:=
integral_closure(F,S,
2,
[[7,4]],
[S|],
S.1^4+S.2^7+S.1^3*S.2+S.2^6+S.1*S.2^4+S.1^3+S.1*S.2^2);

AssignNames(~integral_closure_S,
["y" cat &cat
[ "_" cat IntegerToString(WT_MATRIX_icS[i,j]) 
: i in [1..#WT_MATRIX_icS]]
: j in [1..#WT_MATRIX_icS[1]]]);


"I=";
for i in [1..#Basis(I)] do
   i,Basis(I)[i];
end for;

"delta=",delta;

"phi=";
for i in [1..#WT_MATRIX_icS[1]] do
i,integral_closure_S.i@phi;
end for;

"psi=";
for i in [1..Rank(S)] do
i,S.i@psi;
end for;


//MAGMA's Integral Closure algorithm////////////////////
F:=FiniteField(2);
FF<x>:=FunctionField(F);
P<y>:=PolynomialRing(FF);
f:=y^4+x^7+y^3*x+x^6+y*x^4+y^3+y*x^2;
Ff<Y>:=FunctionField(f);
C<X>:=CoefficientRing(Ff);
INT:=Integers(C);
IC:=IntegralClosure(INT,Ff);
B:=Basis(IC);B;
//gives the result
//[ 1, 
//  Y, 
//  1/X*Y^2 + 1/X*Y, 
//  1/(X^5 + X^4 + X^3)*Y^3 + (X + 1)/X^3*Y^2 + (X + 1)/X^2*Y + 1/(X^2 + X + 1) ]
// elements of orders 0,7,10, and 6 respectively, with B[3]-x*B[4]-B[2] (?) of order 5/////////////////


//SINGULAR's normal function would give the following/////////////
// LIB "normal.lib";
// ring r=2,(y,x),wp(7,4);
// ideal i=y4+x7+y3x+x6+yx4+y3+yx2;
// list nor=normal(i);
// def R=nor[1];
// setring R;
// normap;//corresponding to my psi///////////////////////
////////////////////normap[1]=T(1)*T(3)+T(2)
////////////////////normap[2]=T(1)
// norid;// a lex basis in T(1)=x,T(2), T(3), T(4), whatever they are//////////////////////////////////
///////norid[1]=T(1)^2+T(1)*T(2)+T(2)^2+T(3)^2+T(1)*T(4)+T(4)^2+T(1)+T(2)+T(4)
///////norid[2]=T(1)*T(2)*T(3)+T(2)^2*T(3)+T(3)^3+T(1)*T(3)*T(4)+T(3)*T(4)^2+T(3)^2+T(1)*T(4)+T(1)+T(4)+1
///////norid[3]=T(2)^2*T(3)+T(1)*T(3)^2+T(2)*T(3)^2+T(1)*T(3)*T(4)+T(3)^2*T(4)+T(3)*T(4)^2+T(1)*T(2)+T(2)^2+T(1)*T(3)+T(3)^2+T(1)*T(4)+T(2)*T(4)+T(3)*T(4)+T(4)^2+T(1)+1
///////norid[4]=T(2)^3+T(1)*T(3)^2+T(2)*T(3)^2+T(2)^2*T(4)+T(3)^2*T(4)+T(2)*T(4)^2+T(4)^3+T(2)^2+T(1)*T(3)+T(3)^2+T(4)^2+T(3)
///////norid[5]=T(1)*T(2)^2+T(1)*T(4)^2+T(1)*T(2)+T(3)^2+T(3)*T(4)+T(1)+T(4)+1