Cayley Hypersurfaces

This is the Cayley surface when N = 3. The next few are as follows. x4 = x1x3 + 1 2 x2 2 − x1 x2 + 1 4 x1 4 x5 = x1x4 + x2x3 − x1 x3 − x1x2 2 + x1 x2 − 1 5 x1 5 x6 = x1x5 + x2x4 + 1 2 x3 2 − x1 x4 − 2x1x2x3 − 1 3 x2 3 + x1 x3 + 3 2 x1 x2 2 − x1 x2 + 1 6 x1 . Since the first term in (1) is −xN and this is the only occurrence of this variable, these hypersurfaces are polynomial graphs over the remaining variables. The Cayley surface is affine homogeneous. This follows immediately from Φ3 being annihilated by the following two linearly independent affine vector fields: ∂ ∂x1 + x1 ∂ ∂x2 + x2 ∂ ∂x3 and ∂ ∂x2 + x1 ∂ ∂x3 .