tailieunhanh - Báo cáo toán học: "n the non–existence of certain hyperovals in dual Andr´ planes of order 22h e"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: On the non–existence of certain hyperovals in dual Andr´ planes of order 22h e. | On the non-existence of certain hyperovals in dual André planes of order 22h Angela Aguglia Dipartimento di Matematica Politecnico di Bari Via Orabona 4 I-70125 Bari Italy Luca Giuzzi Dipartimento di Matematica Facolta di Ingegneria Università degli Studi di Brescia Via Valotti 9 I-25133 Brescia Italy giuzzi@ Submitted Jul 31 2008 Accepted Oct 13 2008 Published Oct 20 2008 Mathematics Subject Classification 51E15 51E21 Abstract No regular hyperoval of the Desarguesian affine plane AG 2 22h with h 1 is inherited by a dual Andre plane of order 22h and dimension 2 over its kernel. 1 Introduction The general question on existence of ovals in finite non-Desarguesian planes is still open and appears to be difficult. It has been shown by computer search that there exist some planes of order 16 without ovals see 11 . On the other hand ovals have been constructed in several finite planes one of the most fruitful approaches in this search has been that of inherited oval due to Korchmáros 5 6 . Korchmáros idea relies on the fact that any two planes and 2 of the same order have the same number of points and lines thus their point sets as well as some lines may be identified. If e is an oval of it might happen that e regarded as a point set turns also out to be an oval of 2 although and 2 differ in some in general several point-line incidences in this case e is called an inherited oval of 2 from see also 2 Page 728 . In practice it is usually convenient to take to be the Desarguesian affine plane AG 2 q of order a prime power q. The case in which 2 is the Hall plane H q2 of order q2 was investigated in 5 and inherited ovals were found. For q odd this also proves the Research supported by the Italian Ministry MIUR Strutture geometriche combinatoria e loro ap-plicazioni. THE ELECTRONIC JOURNAL OF COMBINATORICS 15 2008 N37 1 existence of inherited ovals in the dual plane of H q2 which is a Moulton plane M q2 of the same order. Moulton planes have .