I am attempting to take the projective closure of the intersection of the following affine polynomials (not the intersection of the closure!):
$$y^3-y-x^2= 0 $$ $$w^3-w+y^7-y^5-x^4y^3+x^4y = 0.$$
This affine intersection is a curve of dimension one. Unfortunately, I haven't been able to enter this lovely affine curve defined by this intersection into sage. When I attempt as follows:
A.<x,y,w> = AffineSpace(QQ, 3)
P.<u,v,t,s>=ProjectiveSpace(QQ,3)
C = Curve([y^3-y-x^2, w^3-w+y^7-y^5-x^4*y^3+x^4*y], A)
D=C.projective_closure(1,P)
I get an error at the definition of C due to the second polynomial:
TypeError: F (=[-x^4*y^3 + y^7 + x^4*y - y^5 + w^3 - w]) must be a list or tuple of polynomials of the coordinate ring of A (=Affine Space of dimension 3 over Finite Field of size 3)
I am so confused because this is absolutely in the coordinate ring of $A$. Why am I getting this type error? How can I enter this affine intersection into sage, so that I may take its closure?
As an aside, I can enter the intersection of the closure, which is not what I want, as follows:
x,y,z,w = GF(3)['x,y,z,w'].gens()
C = Curve([y^3-y*z^2-x^2*z, w^3*z^4-w*z^6+y^7-y^5*z^2-x^4*y^3+x^4*y*z^2]); C
The intersection of the closure has an extra irreducible component $[x: 0: w: 0]$,
