-- run M2
-- copy the code below
--
-- If you have Macaulay2 in your PC, just:
-- download the file "Implicitization.m2" from
-- http://mate.dm.uba.ar/~nbotbol/Implicitization.m2
-- run M2
-- copy the code below

-- Exercise code

 S  =  QQ[s,t];
f0  =  2+s^2*t^6
f1  =  s*t^6+2
f2  =  s*t^5-3*s*t^3
f3  =  s*t^4+5*s^2*t^6

needsPackage "Polyhedra";
load "Implicitization.m2";
L={f0,f1,f2,f3};

----- First: with the N(f) toric embedding

latticePoints polynomialsToPolytope L
tEmb=newToricEmbedding L
A= teToricRing tEmb
describe A
g = teToricRationalMap L;
rM=representationMatrix (teToricRationalMap L,2);
R=QQ[s,t,X_0..X_3];
Eq=sub(sub(rM,R), {X_0=>(sub(f0,R)), X_1=>(sub(f1,R)), X_2=>(sub(f2,R)), X_3=>(sub(f3,R))});
rank rM
rank Eq


iEq = implicitEq (L,2)
(factor iEq)#2#0

----- Second: with the triangle that gives the P^2 compactification

Q2=convexHull matrix{{0,0,1},{0,1,0}}
gQ2 = teToricRationalMap (L,Q2)
rMQ2=representationMatrix (gQ2,14);
R=QQ[s,t,X_0..X_3];
EqQ2=sub(sub(rMQ2,R), {X_0=>(sub(f0,R)), X_1=>(sub(f1,R)), X_2=>(sub(f2,R)), X_3=>(sub(f3,R))});
rank rMQ2
rank EqQ2


-- Optional
----- Third: try with the 1 by 1 rectangle that gives a segre-veronese embedding of P^1xP^1

Q3=convexHull matrix{{0,0,1,1},{0,1,0,1}}
gQ3 = teToricRationalMap (L,Q3)
rMQ3=representationMatrix (gQ3,5);
R=QQ[s,t,X_0..X_3];
EqQ3=sub(sub(rMQ3,R), {X_0=>(sub(f0,R)), X_1=>(sub(f1,R)), X_2=>(sub(f2,R)), X_3=>(sub(f3,R))});
rank rMQ3
rank EqQ3

iEq3 = implicitEq (L,2,Q3)

-- Optional
----- Third: try with this other triangle. Works better?

Q1=convexHull matrix{{0,0,1},{0,3,3}}
gQ1 = teToricRationalMap (L,Q1)
rMQ1=representationMatrix (gQ1,2);
R=QQ[s,t,X_0..X_3];
EqQ1=sub(sub(rMQ1,R), {X_0=>(sub(f0,R)), X_1=>(sub(f1,R)), X_2=>(sub(f2,R)), X_3=>(sub(f3,R))});
rank rMQ1
rank EqQ1

 iEq1 = implicitEq (L,2,Q1)
(factor iEq1)#2#0