So I am reading through William Fulton's Algebraic Curves and currently I am reading that bit about projective curves. Now, when the author discussed about the affine case (in the earlier chapters 1-3), for example, to define affine plane curves in Chapter 3, and all the discussion following he considered $F$ to be a polynomial.
In Chapter 2, he talked about how we can write a polynomial of n variables as a sum of forms and gave a few results related to them. Then when Projective varieties were introduced, he again gave a few result related to forms and from that point onwards, whenever a polynomial $F \in k(\mathbb{P^2}) $ was given it's always assumed to be a form.
Now, what I don't understand is, why is it sufficient to just consider forms in the projective case in contrary to considering polynomial(a sum of forms) in the affine case?
I would appreciate any kind of help/hint(s).
Thanks a lot!
In projective space, everything is only defined up to scalar multiples. So if you had a non homogeneous polynomial, the question of whether or not a polynomial vanishes at a point of projective space isn't even well defined. For instance, $(0,1,0)$ is a solution to $x + y = 1$. This is the same point in the projective plane as $(0,2,0)$, which is not a solution.