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α-Flocks with Oval Herds and Monomial Hyperovals

✍ Scribed by W.E. Cherowitzo; L. Storme

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
337 KB
Volume
4
Category
Article
ISSN
1071-5797

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✦ Synopsis

In PG (3, q), q even, Cherowitzo made a detailed study of flocks of a cone with a translation oval as base; also called -flocks . To a flock of a quadratic cone in PG(3, q), q even, there always corresponds a set of q#1 ovals in PG(2, q), called an oval herd. To an -flock of a cone with an arbitrary translation oval as base, there corresponds a herd of q#1 permutation polynomials. For some, but not for all, known examples of -flocks, these q#1 permutation polynomials define an oval herd. This leads to the fundamental problem of determining which -flocks correspond to an oval herd. This article studies a class of -flocks and explicitly describes which members of this class have an associated oval herd. To achieve this goal, all monomial hyperovals +(1, t, tI)"" t3F O ,6+(0, 1, 0), (0, 0, 1), with k"2G#2H, iOj, are determined.

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Monomial Flocks and Herds Containing a M
Monomial Flocks and Herds Containing a Monomial Oval
✍ Tim Penttila; L Storme 📂 Article 📅 1998 🏛 Elsevier Science 🌐 English ⚖ 367 KB

Let F be a flock of the quadratic cone Q: X 2 2 =X 1 X 3 , in PG(3, q), q even, and let 6 t : X 0 = x t X 1 + t 1Â2 X 2 + z t X 3 , t # F q , be the q planes defining the flock F. A flock is equivalent to a herd of ovals in PG(2, q), q even, and to a flock generalized quadrangle of order (q 2 , q).