Dimensional dual hyperovals in projective spaces and

The two subjects of the title are studied, first independently and then by making them interact. Many questions arise from this interaction. In our intrinsic study of the first subject, we construct a new family of examples and define the notions of quotient and universality. For the second subject, we restrict the class c·AG∗ of circular extensions of dual affine spaces under the two geometrical conditions (LL) and (T) by showing that, apart from some extreme cases, every such rank three geometry can be erected into a rank 4 geometry. In particular, c·AGn∗(q)-geometries satisfying (LL) and (T) with n⩾3 must have q even. We then deduce similar results for the first subject