project elliptope

# elliptope R.<w,x,y,z,a,c0,c1,c2,c3> = PolynomialRing(QQ,order="lex") M = Matrix(3,3,[w,x,y,x,w,z,y,z,w]) f = det(M) f 

w^3 - w*x^2 - w*y^2 - w*z^2 + 2*x*y*z

w^3 - w*x^2 - w*y^2 - w*z^2 + 2*x*y*z

# Dual of elliptope I = R.ideal([f,c0-a*f.derivative(w),c1-a*f.derivative(x),c2-a*f.derivative(y),c3-a*f.derivative(z)]) gb =I.groebner_basis() g = gb[-1] g 

c0*c1*c2*c3 - 1/2*c1^2*c2^2 - 1/2*c1^2*c3^2 - 1/2*c2^2*c3^2

c0*c1*c2*c3 - 1/2*c1^2*c2^2 - 1/2*c1^2*c3^2 - 1/2*c2^2*c3^2

# Singular points of dual intersected with plane c1 = a0c0 + a2c2 + a3c3. # i.e. singular points of X* cap F* in our terminology var('b0,b2,b3') var('a0,a2,a3') h=g(c0=b0,c1=a0*b0+a2*b2+a3*b3,c2=b2,c3=b3) derivs = [h.derivative(v) for v in [b0,b2,b3]] solve([fn == 0 for fn in derivs],b0,b2,b3) 

[[b0 == r1, b2 == -a0*r1/a2, b3 == 0], [b0 == r2, b2 == 0, b3 ==
-a0*r2/a3], [b0 == r3, b2 == 0, b3 == 0]]

[[b0 == r1, b2 == -a0*r1/a2, b3 == 0], [b0 == r2, b2 == 0, b3 == -a0*r2/a3], [b0 == r3, b2 == 0, b3 == 0]]

# So singular points are (1,-a0/a2,0); (1,0,-a0/a3); and (1,0,0). # Or (a2,-a0,0), (a3,0,-a0), (1,0,0). # Now I'll try to look locally at these points. # I am pretty unsure how to do this. 

# Looking locally at (1,0,0). # ???