restart
loadPackage "EliminationMatrices"
--help detComplex
R=QQ[a,b,c,d,e,f,x,y]
f1 = a*x^2+b*x*y+c*y^2
f2 = 2*a*x+b*y
--f2 = d*x^2+e*x*y+f*y^2
M = matrix{{f1,f2}}
l = {x,y}
K=koszul M -- the Koszul complex
K.dd
nu=5 -- graded component
mapsComplex(nu,l,koszul M) -- maps of the complex in degree nu 
minorsComplex(nu,l,koszul M) -- maps of the complex in degree nu 
L=listDetComplex(nu,l,koszul M) -- decomposition in minors 
detComplex (nu,l,koszul M) -- graded component of degree 2

(A,B)=macaulayFormula(l,M)
det A / det B

---------------- 

R=QQ[a,b,c,d,e,f,g,h,i,x,y,z] 
f1 = a*x+b*y+c*z 
f2 = d*x+e*y+f*z 
--f3 = g*x+h*y+i*z 
f3 = g*x^2+h*y^2+i*z^2 
M = matrix{{f1,f2,f3}}
l = {x,y,z}
K=koszul M -- the Koszul complex
K.dd
nu=5 -- graded component
mapsComplex(nu,l,koszul M) -- maps of the complex in degree nu 
minorsComplex(nu,l,koszul M) -- maps of the complex in degree nu 
L=listDetComplex(nu,l,koszul M) -- decomposition in minors 
detComplex (nu,l,koszul M) -- graded component of degree 2

(A,B)=macaulayFormula(l,M)
det A / det B


----------------  
restart
loadPackage "EliminationMatrices"
R=QQ[z,x0,x1,x2,T0,T1,T2,T3,MonomialOrder=>Lex]
-- Sphere parameterization
f0=x0^2+x1^2+x2^2
f1=x0^2-x1^2-x2^2
f2=2*x0*x1
f3=2*x0*x2
f1^2+f2^2+f3^2-f0^2 -- expected equation: T1^2+T2^2+T3^2-T0^1
F=matrix{{f0,f1,f2,f3}}

I=ideal F
primaryDecomposition saturate(I,ideal(x0,x1,x2)) - 2 base points

-- Resultant
l={x0,x1,x2}
eqF=matrix{{T0*f1-T1*f0,T0*f2-T2*f0,T0*f3-T3*f0}}
(A,B)=macaulayFormula({x0,x1,x2},eqF)
B
det B
det A

g0=f0 
g1=f1+3*x0*x1
g2=f2 + 5*x1*x2
g3=f3
eqF=matrix{{T0*g1-T1*g0,T0*g2-T2*g0,T0*g3-T3*g0}}
(A,B)=macaulayFormula({x0,x1,x2},eqF)
B
det B
det A
Eq1=det A / det B
factor Eq1

Eq2=detComplex (4,l,koszul eqF) -- graded component of degree 2
Eq1+Eq2 
listM=minorsComplex (6,l,koszul eqF)
listM_2

-- Come back to the sphere example

-- Equations of Rees algebra
J=ideal (T0-f0*z,T1-f1*z,T2-f2*z,T3-f3*z)
transpose gens gb J

-- Now, syzygies!
SM=syz F
F
F*SM
I= ideal F
saturate(I,ideal(x0,x1,x2))
(res I).dd--be carefull with reordering
ML=matrix{{T0,T1,T2,T3}}*SM -- generators of Symmetric algebra
IM=ML//matrix{{x0,x1,x2}}
factor det IM_{0,1,2}
factor det IM_{1,2,3}