///////////////////////////////////
// Projections to zy-coordinates //
///////////////////////////////////

LIB "all.lib";
LIB "poly.lib";
LIB "tropical.lib";
LIB "elim.lib";

ring rr = (0,b2,b3,b4,b5), (x,y,z),dp;
poly f=y^2-x*(x-b2^2)*(x-b3^2)*(x-b4^2)*(x+b5^2);
// f;
// -x^5+(b2^2+b3^2+b4^2-b5^2)*x^4+(-b2^2*b3^2-b2^2*b4^2+b2^2*b5^2-b3^2*b4^2+b3^2*b5^2+b4^2*b5^2)*x^3+(b2^2*b3^2*b4^2-b2^2*b3^2*b5^2-b2^2*b4^2*b5^2-b3^2*b4^2*b5^2)*x^2+y^2+(b2^2*b3^2*b4^2*b5^2)*x

ideal I = (f,z-y+b3*b4*b5*x-b5*x^2);

poly fnox= eliminate(I,x)[1];


// Replace z by y and y by x so that the pictures are not flipped.
ring r2 = (0,b2, b3, b4, b5),(x,y),dp;
setring(r2);
map PP = rr,0,x,y;
poly newfnox = PP(fnox);
// newfnox;
// x^5-5*x^4*y+10*x^3*y^2-10*x^2*y^3+5*x*y^4-y^5+(b2^4*b5-b2^2*b3*b4*b5+2*b2^2*b5^3+b3^4*b5-b3^3*b4*b5+2*b3^2*b5^3-b3*b4^3*b5-4*b3*b4*b5^3+b4^4*b5+2*b4^2*b5^3)*x^4+(-4*b2^4*b5+4*b2^2*b3*b4*b5-4*b2^2*b5^3-4*b3^4*b5+4*b3^3*b4*b5-4*b3^2*b5^3+4*b3*b4^3*b5+6*b3*b4*b5^3-4*b4^4*b5-4*b4^2*b5^3)*x^3*y+(6*b2^4*b5-6*b2^2*b3*b4*b5+2*b2^2*b5^3+6*b3^4*b5-6*b3^3*b4*b5+2*b3^2*b5^3-6*b3*b4^3*b5+b3*b4*b5^3+6*b4^4*b5+2*b4^2*b5^3+4*b5^5)*x^2*y^2+(-4*b2^4*b5+4*b2^2*b3*b4*b5-4*b3^4*b5+4*b3^3*b4*b5+4*b3*b4^3*b5-4*b3*b4*b5^3-4*b4^4*b5-4*b5^5)*x*y^3+(b2^4*b5-b2^2*b3*b4*b5+b3^4*b5-b3^3*b4*b5-b3*b4^3*b5+b3*b4*b5^3+b4^4*b5+b5^5)*y^4+(b2^4*b3^4*b5^2-b2^4*b3^3*b4*b5^2-b2^4*b3*b4^3*b5^2+b2^4*b3*b4*b5^4+b2^4*b4^4*b5^2+b2^4*b5^6-b2^2*b3^5*b4*b5^2+b2^2*b3^4*b4^2*b5^2+3*b2^2*b3^3*b4*b5^4+b2^2*b3^2*b4^4*b5^2-7*b2^2*b3^2*b4^2*b5^4+2*b2^2*b3^2*b5^6-b2^2*b3*b4^5*b5^2+3*b2^2*b3*b4^3*b5^4-4*b2^2*b3*b4*b5^6+2*b2^2*b4^2*b5^6-b3^5*b4^3*b5^2+b3^5*b4*b5^4+2*b3^4*b4^4*b5^2-5*b3^4*b4^2*b5^4+b3^4*b5^6-b3^3*b4^5*b5^2+8*b3^3*b4^3*b5^4-4*b3^3*b4*b5^6-5*b3^2*b4^4*b5^4+6*b3^2*b4^2*b5^6+b3*b4^5*b5^4-4*b3*b4^3*b5^6+b4^4*b5^6)*x^3+(-3*b2^4*b3^4*b5^2+3*b2^4*b3^3*b4*b5^2+3*b2^4*b3*b4^3*b5^2-3*b2^4*b3*b4*b5^4-3*b2^4*b4^4*b5^2-3*b2^4*b5^6+3*b2^2*b3^5*b4*b5^2-3*b2^2*b3^4*b4^2*b5^2-3*b2^2*b3^3*b4*b5^4-3*b2^2*b3^2*b4^4*b5^2+9*b2^2*b3^2*b4^2*b5^4-2*b2^2*b3^2*b5^6+3*b2^2*b3*b4^5*b5^2-3*b2^2*b3*b4^3*b5^4+6*b2^2*b3*b4*b5^6-2*b2^2*b4^2*b5^6+3*b3^5*b4^3*b5^2-3*b3^5*b4*b5^4-6*b3^4*b4^4*b5^2+7*b3^4*b4^2*b5^4-3*b3^4*b5^6+3*b3^3*b4^5*b5^2-8*b3^3*b4^3*b5^4+6*b3^3*b4*b5^6+7*b3^2*b4^4*b5^4-6*b3^2*b4^2*b5^6-3*b3*b4^5*b5^4+6*b3*b4^3*b5^6-3*b4^4*b5^6)*x^2*y+(3*b2^4*b3^4*b5^2-3*b2^4*b3^3*b4*b5^2-3*b2^4*b3*b4^3*b5^2+3*b2^4*b3*b4*b5^4+3*b2^4*b4^4*b5^2+3*b2^4*b5^6-3*b2^2*b3^5*b4*b5^2+3*b2^2*b3^4*b4^2*b5^2+3*b2^2*b3^2*b4^4*b5^2-3*b2^2*b3^2*b4^2*b5^4-3*b2^2*b3*b4^5*b5^2-3*b2^2*b3*b4*b5^6-3*b3^5*b4^3*b5^2+3*b3^5*b4*b5^4+6*b3^4*b4^4*b5^2-3*b3^4*b4^2*b5^4+3*b3^4*b5^6-3*b3^3*b4^5*b5^2-3*b3^3*b4*b5^6-3*b3^2*b4^4*b5^4+3*b3*b4^5*b5^4-3*b3*b4^3*b5^6+3*b4^4*b5^6)*x*y^2+(-b2^4*b3^4*b5^2+b2^4*b3^3*b4*b5^2+b2^4*b3*b4^3*b5^2-b2^4*b3*b4*b5^4-b2^4*b4^4*b5^2-b2^4*b5^6+b2^2*b3^5*b4*b5^2-b2^2*b3^4*b4^2*b5^2-b2^2*b3^2*b4^4*b5^2+b2^2*b3^2*b4^2*b5^4+b2^2*b3*b4^5*b5^2+b2^2*b3*b4*b5^6+b3^5*b4^3*b5^2-b3^5*b4*b5^4-2*b3^4*b4^4*b5^2+b3^4*b4^2*b5^4-b3^4*b5^6+b3^3*b4^5*b5^2+b3^3*b4*b5^6+b3^2*b4^4*b5^4-b3*b4^5*b5^4+b3*b4^3*b5^6-b4^4*b5^6)*y^3+(-b2^4*b3^5*b4^3*b5^3+b2^4*b3^5*b4*b5^5+2*b2^4*b3^4*b4^4*b5^3-b2^4*b3^4*b4^2*b5^5+b2^4*b3^4*b5^7-b2^4*b3^3*b4^5*b5^3-b2^4*b3^3*b4*b5^7-b2^4*b3^2*b4^4*b5^5+b2^4*b3*b4^5*b5^5-b2^4*b3*b4^3*b5^7+b2^4*b4^4*b5^7+b2^2*b3^6*b4^4*b5^3-b2^2*b3^6*b4^2*b5^5-2*b2^2*b3^5*b4^5*b5^3-b2^2*b3^5*b4*b5^7+b2^2*b3^4*b4^6*b5^3+2*b2^2*b3^4*b4^4*b5^5+2*b2^2*b3^3*b4^3*b5^7-b2^2*b3^2*b4^6*b5^5-b2^2*b3*b4^5*b5^7)*x^2+(2*b2^4*b3^5*b4^3*b5^3-2*b2^4*b3^5*b4*b5^5-4*b2^4*b3^4*b4^4*b5^3+2*b2^4*b3^4*b4^2*b5^5-2*b2^4*b3^4*b5^7+2*b2^4*b3^3*b4^5*b5^3+2*b2^4*b3^3*b4*b5^7+2*b2^4*b3^2*b4^4*b5^5-2*b2^4*b3*b4^5*b5^5+2*b2^4*b3*b4^3*b5^7-2*b2^4*b4^4*b5^7-2*b2^2*b3^6*b4^4*b5^3+2*b2^2*b3^6*b4^2*b5^5+4*b2^2*b3^5*b4^5*b5^3-2*b2^2*b3^5*b4^3*b5^5+2*b2^2*b3^5*b4*b5^7-2*b2^2*b3^4*b4^6*b5^3-2*b2^2*b3^4*b4^2*b5^7-2*b2^2*b3^3*b4^5*b5^5+2*b2^2*b3^2*b4^6*b5^5-2*b2^2*b3^2*b4^4*b5^7+2*b2^2*b3*b4^5*b5^7+2*b3^6*b4^4*b5^5-4*b3^5*b4^5*b5^5+2*b3^5*b4^3*b5^7+2*b3^4*b4^6*b5^5-4*b3^4*b4^4*b5^7+2*b3^3*b4^5*b5^7)*x*y+(-b2^4*b3^5*b4^3*b5^3+b2^4*b3^5*b4*b5^5+2*b2^4*b3^4*b4^4*b5^3-b2^4*b3^4*b4^2*b5^5+b2^4*b3^4*b5^7-b2^4*b3^3*b4^5*b5^3-b2^4*b3^3*b4*b5^7-b2^4*b3^2*b4^4*b5^5+b2^4*b3*b4^5*b5^5-b2^4*b3*b4^3*b5^7+b2^4*b4^4*b5^7+b2^2*b3^6*b4^4*b5^3-b2^2*b3^6*b4^2*b5^5-2*b2^2*b3^5*b4^5*b5^3+b2^2*b3^5*b4^3*b5^5-b2^2*b3^5*b4*b5^7+b2^2*b3^4*b4^6*b5^3+b2^2*b3^4*b4^2*b5^7+b2^2*b3^3*b4^5*b5^5-b2^2*b3^2*b4^6*b5^5+b2^2*b3^2*b4^4*b5^7-b2^2*b3*b4^5*b5^7-b3^6*b4^4*b5^5+2*b3^5*b4^5*b5^5-b3^5*b4^3*b5^7-b3^4*b4^6*b5^5+2*b3^4*b4^4*b5^7-b3^3*b4^5*b5^7)*y^2+(-b2^4*b3^6*b4^4*b5^6+2*b2^4*b3^5*b4^5*b5^6-b2^4*b3^5*b4^3*b5^8-b2^4*b3^4*b4^6*b5^6+2*b2^4*b3^4*b4^4*b5^8-b2^4*b3^3*b4^5*b5^8+b2^2*b3^7*b4^5*b5^6-2*b2^2*b3^6*b4^6*b5^6+b2^2*b3^6*b4^4*b5^8+b2^2*b3^5*b4^7*b5^6-2*b2^2*b3^5*b4^5*b5^8+b2^2*b3^4*b4^6*b5^8)*x+(b2^4*b3^6*b4^4*b5^6-2*b2^4*b3^5*b4^5*b5^6+b2^4*b3^5*b4^3*b5^8+b2^4*b3^4*b4^6*b5^6-2*b2^4*b3^4*b4^4*b5^8+b2^4*b3^3*b4^5*b5^8-b2^2*b3^7*b4^5*b5^6+2*b2^2*b3^6*b4^6*b5^6-b2^2*b3^6*b4^4*b5^8-b2^2*b3^5*b4^7*b5^6+2*b2^2*b3^5*b4^5*b5^8-b2^2*b3^4*b4^6*b5^8)*y

matrix m = coef(newfnox,xy);
m;
m[1,1]=x^5
m[1,2]=x^4*y
m[1,3]=x^3*y^2
m[1,4]=x^2*y^3
m[1,5]=x*y^4
m[1,6]=y^5
m[1,7]=x^4
m[1,8]=x^3*y
m[1,9]=x^2*y^2
m[1,10]=x*y^3
m[1,11]=y^4
m[1,12]=x^3
m[1,13]=x^2*y
m[1,14]=x*y^2
m[1,15]=y^3
m[1,16]=x^2
m[1,17]=x*y
m[1,18]=y^2
m[1,19]=x
m[1,20]=y
m[2,1]=1
m[2,2]=-5
m[2,3]=10
m[2,4]=-10
m[2,5]=5
m[2,6]=-1
m[2,7]=(b2^4*b5-b2^2*b3*b4*b5+2*b2^2*b5^3+b3^4*b5-b3^3*b4*b5+2*b3^2*b5^3-b3*b4^3*b5-4*b3*b4*b5^3+b4^4*b5+2*b4^2*b5^3)
m[2,8]=(-4*b2^4*b5+4*b2^2*b3*b4*b5-4*b2^2*b5^3-4*b3^4*b5+4*b3^3*b4*b5-4*b3^2*b5^3+4*b3*b4^3*b5+6*b3*b4*b5^3-4*b4^4*b5-4*b4^2*b5^3)
m[2,9]=(6*b2^4*b5-6*b2^2*b3*b4*b5+2*b2^2*b5^3+6*b3^4*b5-6*b3^3*b4*b5+2*b3^2*b5^3-6*b3*b4^3*b5+b3*b4*b5^3+6*b4^4*b5+2*b4^2*b5^3+4*b5^5)
m[2,10]=(-4*b2^4*b5+4*b2^2*b3*b4*b5-4*b3^4*b5+4*b3^3*b4*b5+4*b3*b4^3*b5-4*b3*b4*b5^3-4*b4^4*b5-4*b5^5)
m[2,11]=(b2^4*b5-b2^2*b3*b4*b5+b3^4*b5-b3^3*b4*b5-b3*b4^3*b5+b3*b4*b5^3+b4^4*b5+b5^5)
m[2,12]=(b2^4*b3^4*b5^2-b2^4*b3^3*b4*b5^2-b2^4*b3*b4^3*b5^2+b2^4*b3*b4*b5^4+b2^4*b4^4*b5^2+b2^4*b5^6-b2^2*b3^5*b4*b5^2+b2^2*b3^4*b4^2*b5^2+3*b2^2*b3^3*b4*b5^4+b2^2*b3^2*b4^4*b5^2-7*b2^2*b3^2*b4^2*b5^4+2*b2^2*b3^2*b5^6-b2^2*b3*b4^5*b5^2+3*b2^2*b3*b4^3*b5^4-4*b2^2*b3*b4*b5^6+2*b2^2*b4^2*b5^6-b3^5*b4^3*b5^2+b3^5*b4*b5^4+2*b3^4*b4^4*b5^2-5*b3^4*b4^2*b5^4+b3^4*b5^6-b3^3*b4^5*b5^2+8*b3^3*b4^3*b5^4-4*b3^3*b4*b5^6-5*b3^2*b4^4*b5^4+6*b3^2*b4^2*b5^6+b3*b4^5*b5^4-4*b3*b4^3*b5^6+b4^4*b5^6)
m[2,13]=(-3*b2^4*b3^4*b5^2+3*b2^4*b3^3*b4*b5^2+3*b2^4*b3*b4^3*b5^2-3*b2^4*b3*b4*b5^4-3*b2^4*b4^4*b5^2-3*b2^4*b5^6+3*b2^2*b3^5*b4*b5^2-3*b2^2*b3^4*b4^2*b5^2-3*b2^2*b3^3*b4*b5^4-3*b2^2*b3^2*b4^4*b5^2+9*b2^2*b3^2*b4^2*b5^4-2*b2^2*b3^2*b5^6+3*b2^2*b3*b4^5*b5^2-3*b2^2*b3*b4^3*b5^4+6*b2^2*b3*b4*b5^6-2*b2^2*b4^2*b5^6+3*b3^5*b4^3*b5^2-3*b3^5*b4*b5^4-6*b3^4*b4^4*b5^2+7*b3^4*b4^2*b5^4-3*b3^4*b5^6+3*b3^3*b4^5*b5^2-8*b3^3*b4^3*b5^4+6*b3^3*b4*b5^6+7*b3^2*b4^4*b5^4-6*b3^2*b4^2*b5^6-3*b3*b4^5*b5^4+6*b3*b4^3*b5^6-3*b4^4*b5^6)
m[2,14]=(3*b2^4*b3^4*b5^2-3*b2^4*b3^3*b4*b5^2-3*b2^4*b3*b4^3*b5^2+3*b2^4*b3*b4*b5^4+3*b2^4*b4^4*b5^2+3*b2^4*b5^6-3*b2^2*b3^5*b4*b5^2+3*b2^2*b3^4*b4^2*b5^2+3*b2^2*b3^2*b4^4*b5^2-3*b2^2*b3^2*b4^2*b5^4-3*b2^2*b3*b4^5*b5^2-3*b2^2*b3*b4*b5^6-3*b3^5*b4^3*b5^2+3*b3^5*b4*b5^4+6*b3^4*b4^4*b5^2-3*b3^4*b4^2*b5^4+3*b3^4*b5^6-3*b3^3*b4^5*b5^2-3*b3^3*b4*b5^6-3*b3^2*b4^4*b5^4+3*b3*b4^5*b5^4-3*b3*b4^3*b5^6+3*b4^4*b5^6)
m[2,15]=(-b2^4*b3^4*b5^2+b2^4*b3^3*b4*b5^2+b2^4*b3*b4^3*b5^2-b2^4*b3*b4*b5^4-b2^4*b4^4*b5^2-b2^4*b5^6+b2^2*b3^5*b4*b5^2-b2^2*b3^4*b4^2*b5^2-b2^2*b3^2*b4^4*b5^2+b2^2*b3^2*b4^2*b5^4+b2^2*b3*b4^5*b5^2+b2^2*b3*b4*b5^6+b3^5*b4^3*b5^2-b3^5*b4*b5^4-2*b3^4*b4^4*b5^2+b3^4*b4^2*b5^4-b3^4*b5^6+b3^3*b4^5*b5^2+b3^3*b4*b5^6+b3^2*b4^4*b5^4-b3*b4^5*b5^4+b3*b4^3*b5^6-b4^4*b5^6)
m[2,16]=(-b2^4*b3^5*b4^3*b5^3+b2^4*b3^5*b4*b5^5+2*b2^4*b3^4*b4^4*b5^3-b2^4*b3^4*b4^2*b5^5+b2^4*b3^4*b5^7-b2^4*b3^3*b4^5*b5^3-b2^4*b3^3*b4*b5^7-b2^4*b3^2*b4^4*b5^5+b2^4*b3*b4^5*b5^5-b2^4*b3*b4^3*b5^7+b2^4*b4^4*b5^7+b2^2*b3^6*b4^4*b5^3-b2^2*b3^6*b4^2*b5^5-2*b2^2*b3^5*b4^5*b5^3-b2^2*b3^5*b4*b5^7+b2^2*b3^4*b4^6*b5^3+2*b2^2*b3^4*b4^4*b5^5+2*b2^2*b3^3*b4^3*b5^7-b2^2*b3^2*b4^6*b5^5-b2^2*b3*b4^5*b5^7)
m[2,17]=(2*b2^4*b3^5*b4^3*b5^3-2*b2^4*b3^5*b4*b5^5-4*b2^4*b3^4*b4^4*b5^3+2*b2^4*b3^4*b4^2*b5^5-2*b2^4*b3^4*b5^7+2*b2^4*b3^3*b4^5*b5^3+2*b2^4*b3^3*b4*b5^7+2*b2^4*b3^2*b4^4*b5^5-2*b2^4*b3*b4^5*b5^5+2*b2^4*b3*b4^3*b5^7-2*b2^4*b4^4*b5^7-2*b2^2*b3^6*b4^4*b5^3+2*b2^2*b3^6*b4^2*b5^5+4*b2^2*b3^5*b4^5*b5^3-2*b2^2*b3^5*b4^3*b5^5+2*b2^2*b3^5*b4*b5^7-2*b2^2*b3^4*b4^6*b5^3-2*b2^2*b3^4*b4^2*b5^7-2*b2^2*b3^3*b4^5*b5^5+2*b2^2*b3^2*b4^6*b5^5-2*b2^2*b3^2*b4^4*b5^7+2*b2^2*b3*b4^5*b5^7+2*b3^6*b4^4*b5^5-4*b3^5*b4^5*b5^5+2*b3^5*b4^3*b5^7+2*b3^4*b4^6*b5^5-4*b3^4*b4^4*b5^7+2*b3^3*b4^5*b5^7)
m[2,18]=(-b2^4*b3^5*b4^3*b5^3+b2^4*b3^5*b4*b5^5+2*b2^4*b3^4*b4^4*b5^3-b2^4*b3^4*b4^2*b5^5+b2^4*b3^4*b5^7-b2^4*b3^3*b4^5*b5^3-b2^4*b3^3*b4*b5^7-b2^4*b3^2*b4^4*b5^5+b2^4*b3*b4^5*b5^5-b2^4*b3*b4^3*b5^7+b2^4*b4^4*b5^7+b2^2*b3^6*b4^4*b5^3-b2^2*b3^6*b4^2*b5^5-2*b2^2*b3^5*b4^5*b5^3+b2^2*b3^5*b4^3*b5^5-b2^2*b3^5*b4*b5^7+b2^2*b3^4*b4^6*b5^3+b2^2*b3^4*b4^2*b5^7+b2^2*b3^3*b4^5*b5^5-b2^2*b3^2*b4^6*b5^5+b2^2*b3^2*b4^4*b5^7-b2^2*b3*b4^5*b5^7-b3^6*b4^4*b5^5+2*b3^5*b4^5*b5^5-b3^5*b4^3*b5^7-b3^4*b4^6*b5^5+2*b3^4*b4^4*b5^7-b3^3*b4^5*b5^7)
m[2,19]=(-b2^4*b3^6*b4^4*b5^6+2*b2^4*b3^5*b4^5*b5^6-b2^4*b3^5*b4^3*b5^8-b2^4*b3^4*b4^6*b5^6+2*b2^4*b3^4*b4^4*b5^8-b2^4*b3^3*b4^5*b5^8+b2^2*b3^7*b4^5*b5^6-2*b2^2*b3^6*b4^6*b5^6+b2^2*b3^6*b4^4*b5^8+b2^2*b3^5*b4^7*b5^6-2*b2^2*b3^5*b4^5*b5^8+b2^2*b3^4*b4^6*b5^8)
m[2,20]=(b2^4*b3^6*b4^4*b5^6-2*b2^4*b3^5*b4^5*b5^6+b2^4*b3^5*b4^3*b5^8+b2^4*b3^4*b4^6*b5^6-2*b2^4*b3^4*b4^4*b5^8+b2^4*b3^3*b4^5*b5^8-b2^2*b3^7*b4^5*b5^6+2*b2^2*b3^6*b4^6*b5^6-b2^2*b3^6*b4^4*b5^8-b2^2*b3^5*b4^7*b5^6+2*b2^2*b3^5*b4^5*b5^8-b2^2*b3^4*b4^6*b5^8)

poly m7 = m[2,7];
poly m8 = m[2,8];
poly m9 = m[2,9];
poly m10 = m[2,10];
poly m11 = m[2,11];
poly m12 = m[2,12];
poly m13 = m[2,13];
poly m14 = m[2,14];
poly m15 = m[2,15];
poly m16 = m[2,16];
poly m17 = m[2,17];
poly m18 = m[2,18];
poly m19 = m[2,19];
poly m20 = m[2,20];

// We write the output coefficients:
write(":w coefficientsProjectionyz.txt", "mx4 =", m7, "", "mx3y =", m8, "", "mx2y2 =", m9, "", "mxy3 =", m10, "", "my4 =", m11, "", "mx3 =", m12, "", "mx2y =", m13, "", "mxy2 =", m14, "", "my3 =", m15, "", "mx2 =", m16, "", "mxy =", m17, "", "my2 =", m18, "", "mx =", m19, "", "my =", m20, "");