Well, it's defining it in projective space; in projective space, the polynomial has to be homogeneous to make sense. If you were working in affine space, you w

Well, it's defining it in projective space; in projective space, the polynomial has to be homogeneous to make sense. If you were working in affine space, you wouldn't have that requirement -- but then the theorem wouldn't be true without first passing to projective space, I imagine. Which would involve taking your polynomial and adding an extra variable so as to make it homogeneous.