Disjoint unions of dimensional dual hyperovals

Abstract

The notion of sub-objects and their disjoint union is introduced for a dimensional dual arc. This naturally motivates a problem to decompose a dimensional dual hyperoval (DHO for short) into the disjoint union of some subdual arcs, including a subDHO, because such an expression is useful to calculate its universal cover, as suggested by an elementary observation about its ambient space. Under mild restrictions, a criterion is obtained for a DHO ${\mathcal{ℬ}}_1$ of rank $n$ over two element field to be extended to a DHO $\mathcal{A}$ of rank $n+1$ so that $\mathcal{A}$ is a disjoint union of ${\mathcal{ℬ}}_1$ and some subDHO ${\mathcal{ℬ}}_2$ of rank $n$. Under the choice of a complement to ${\mathcal{ℬ}}_1$, such $\mathcal{A}$ as well as ${\mathcal{ℬ}}_2$ are uniquely determined by ${\mathcal{ℬ}}_1$, if they exist. Several known families of DHOs are examined whether they can be extended to DHOs in the above form, but no example is found unless they are bilinear. If a subDHO ${\mathcal{ℬ}}_1$ is bilinear over a specified complement, the criterion is satisfied, and thus there exists a unique pair $\left(\mathcal{A},{\mathcal{ℬ}}_2\right)$ of DHOs satisfying the above conditions. This makes clear the meaning of the construction called “extension" by Dempwolff and Edel for bilinear DHOs.