#### Vol. 14, No. 1, 2015

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Disjoint unions of dimensional dual hyperovals

### Satoshi Yoshiara

Vol. 14 (2015), No. 1, 43–76
##### 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 ${\mathsc{ℬ}}_{1}$ of rank $n$ over two element field to be extended to a DHO $\mathsc{A}$ of rank $n+1$ so that $\mathsc{A}$ is a disjoint union of ${\mathsc{ℬ}}_{1}$ and some subDHO ${\mathsc{ℬ}}_{2}$ of rank $n$. Under the choice of a complement to ${\mathsc{ℬ}}_{1}$, such $\mathsc{A}$ as well as ${\mathsc{ℬ}}_{2}$ are uniquely determined by ${\mathsc{ℬ}}_{1}$, 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 ${\mathsc{ℬ}}_{1}$ is bilinear over a specified complement, the criterion is satisfied, and thus there exists a unique pair $\left(\mathsc{A},{\mathsc{ℬ}}_{2}\right)$ of DHOs satisfying the above conditions. This makes clear the meaning of the construction called “extension" by Dempwolff and Edel for bilinear DHOs.

##### Mathematical Subject Classification 2010
Primary: 05B25, 51A45, 51E20