The Jamison method in galois geometries
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In a fundamental paper R.E. Jamison showed, among other things, that any subset of the points of AG(n, q) that intersects all hyperplanes contains at least n(q − 1) + 1 points. Here we show that the method of proof used by Jamison can be applied to several other basic problems in finite geometries of a varied nature. These problems include the celebrated flock theorem and also the characterization of the elements of GF(q) as a set of squares in GF(q2) with certain properties. This last result, due to A. Blokhuis, settled a well-known conjecture due to J.H. van Lint and the late J. MacWilliams.
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- Buekenhout, F. 1971. An axiomatic of inversive spaces. J. Combin. Theory Ser. A11:208–212.Google Scholar
- Bruck, R.H. 1973. Circle geometry in higher dimensions II. Geom. Dedicata2:133–188.Google Scholar
- Blokhuis, A. 1984. On subsets of GF(q 2) with square differences, Nederl. Akad. Wetensch. Indag. Math.46:369–372.Google Scholar
- Bruen, A.A. 1975. Subregular spreads and indicator sets. Canad. J. Math. 27: 1141–1148.Google Scholar
- Bruen, A.A. (forthcoming) Intersection sets and polynomial multiplicities over finite fields. J.C.T. Series A.Google Scholar
- Fisher, J.C. 1988. Conics, order and k-arcs in AG(2, q) with q odd. J. of Geom. 32:21–39.Google Scholar
- Fisher, J.C. and Thas, J.A. 1979. Flocks in PG(3, q). Math. Z.169:1–11.Google Scholar
- Jamison, R.E. 1977. Covering finite fields with cosets of subspcaes. J. Combin. Theory Ser. A 22:253–266.Google Scholar
- Orr, W.R. 1976. A characterization of subregular spreads in finite 3-space. Geometrical Dedicata5:43–50.Google Scholar
- Orr, W.F. 1973. The Miquelian inversive plane IP(q) and the associated projective planes. Ph.D. Thesis, Madison.Google Scholar
- Thas, J.A. (forthcoming) Recent results on flocks, maximal exterior sets and inversive planes, Ann. Discrete Math. Google Scholar