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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Translation planes of order $ q\sp{2}$: asymptotic estimates

Author: Gary L. Ebert
Journal: Trans. Amer. Math. Soc. 238 (1978), 301-308
MSC: Primary 05B25; Secondary 50D30
MathSciNet review: 0480096
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Abstract: R. H. Bruck has pointed out the one-to-one correspondence between the isomorphism classes of certain translation planes, called subregular, and the equivalence classes of disjoint circles in a finite miquelian inversive plane $ IP(q)$. The problem of determining the number of isomorphism classes of translation planes is old and difficult. Let q be an odd prime-power. In this paper, a study of sets of disjoint circles in $ IP(q)$ enables the author to find an asymptotic estimate of the number of isomorphism classes of translation planes of order $ {q^2}$ which are subregular of index 3 or 4. It is conjectured (and proved for $ n \leqslant 3$) that, given a set of n disjoint circles in $ IP(q)$, the numbers of circles disjoint from each of the given n circles is asymptotic to $ {q^3}/{2^n}$. This conjecture, if true, would allow one to estimate the number of subregular translation planes of order $ {q^2}$ with any positive index.

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Keywords: Subregular translation planes, finite miquelian inversive planes, disjoint circles, bundle, pencil, inversion, conjugate pairs of points, linear sets of circles, projective linear group, asymptotic estimates
Article copyright: © Copyright 1978 American Mathematical Society