Vol. 14, No. 1, 2015

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Disjoint unions of dimensional dual hyperovals

Satoshi Yoshiara

Vol. 14 (2015), No. 1, 43–76

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 1 of rank n over two element field to be extended to a DHO A of rank n + 1 so that A is a disjoint union of 1 and some subDHO 2 of rank n. Under the choice of a complement to 1, such A as well as 2 are uniquely determined by 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 1 is bilinear over a specified complement, the criterion is satisfied, and thus there exists a unique pair (A,2) 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
Received: 14 October 2013
Satoshi Yoshiara