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Explain why the following polynomials are homogeneous, and find each degree. (1) x2 + y 2 − z 2 (2) xz − y 2 (3) x3 + 3xy 2 + 4y 3 (4) x4 + x2 y 2 28 1. 14. Explain why the following polynomials are not homogeneous. 15. Show that if the homogeneous equation Ax + By + Cz = 0 holds for the point (x, y, z) in C3 − {(0, 0, 0)}, then it holds for every point of C3 that belongs to the equivalence class (x : y : z) in P2 . 16. Show that if the homogeneous equation Ax2 + By 2 + Cz 2 + Dxy + Exz + F yz = 0 holds for the point (x, y, z) in C3 − {(0, 0, 0)}, then it holds for every point of C3 that belongs to the equivalence class (x : y : z) in P2 .

18 1. 16. Suppose ac = 0 (so b = 0). Since either a = 0 or c = 0, we can assume c = 0. Use the real affine transformation x=u+v y= 1−a b u− 1+a b v to transform V(ax2 + bxy + dx + ey + h) to a conic in the uv-plane of the form V(u2 − v 2 + Du + Ev + H). In all three cases we find that the hyperbola can be transformed to V(Au2 − Cv 2 + Du + Ev + H) in the uv-plane, with both A and C positive. We can now complete the transformation of the hyperbola as we did above with parabolas and ellipses. 17. Show that there exist constants R, S and T so that Au2 − Cv 2 + Du + Ev + H = A(u − R)2 − C(v − S)2 − T.

We denote these points as C(Q) = {(x, y) ∈ Q2 : x2 + y 2 = 1}. Recall from the last section, the parameterization ψ : Q → {(x, y) ∈ Q2 : x2 + y 2 = 1} given by λ→ 2λ λ2 − 1 , λ2 + 1 λ2 + 1 . 5. Show that the above map ψ sends Q → C(Q). 9. Degenerate Conics 39 Extend this to a map ψ : P1 (Q) → C(Q) ⊂ P2 (Q) by (λ : μ) → (2λμ : λ2 − μ2 : λ2 + μ2 ), where λ, μ ∈ Z. 5, this gives us a way to produce an infinite number of integer solutions to x2 + y 2 = z 2 . We now want to show that the map ψ is onto, so that we actually obtain all Pythagorean triples.