, Volume 9, Issue 3, pp 315–320

Planar functions over finite fields

  • L. Rónyai
  • T. SzŐnyi


Letp>2 be a prime. A functionf: GF(p)→GF(p) is planar if for everya∃GF(p)*, the functionf(x+a−f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp2, thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8].

We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.

AMS subject classification (1980)

05 B 25 11 T 21 51 E 15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Z. I. Borevich andI. R. Shafarevich, Number theory,Academic Press, New York, 1966.Google Scholar
  2. [2]
    P. Dembowski,Finite Geometries, Springer-Verlag, Berlin, 1968.Google Scholar
  3. [3]
    P. Dembowski andT. G. Ostrom, Planes of ordern with collineation groups of ordern 2,Math. Zeit.,103 (1968), 239–258.CrossRefGoogle Scholar
  4. [4]
    M. Ganley, Polarities of translation planes,Geom. Ded.,1 (1972), 103–116.Google Scholar
  5. [5]
    A. Goncalves andC. Y. Ho, On collineation groups of a projective plane of prime order,Geom. Ded.,20 (1986), 357–366.Google Scholar
  6. [6]
    J. W. P. Hirschfeld,Projective Geometries over Finite Fields, Clarendon Press, Oxford, 1979.Google Scholar
  7. [7]
    K. Ireland andM. Rosen,A classical introduction to modern number theory, GTM 84, Springer-Verlag, Berlin 1982.Google Scholar
  8. [8]
    N. L. Johnson, Projective planes of orderp that admit collineation groups of orderp 2,J. of Geom.,30 (1987), 49–68.CrossRefGoogle Scholar
  9. [9]
    L. Rédei,Lacunary polynomials over finite fields, Akadémiai Kiadó, Budapest, 1973.Google Scholar
  10. [10]
    H. Salzmann, Topological planes,Advances in Math.,2 (1967), 1–60.CrossRefGoogle Scholar
  11. [11]
    B.Segre, Ovals in a finite projective plane,Can. J. Math. (1955), 414–416.Google Scholar
  12. [12]
    B.Segre,Lectures on Modern Geometry, Cremonese Roma, 1961.Google Scholar
  13. [13]
    K. Yamamoto, On Jacobi Sums and Difference Sets,J. Comb. Th. 3 (1967), 146–181.Google Scholar

Copyright information

© Akadémiai Kiadó 1989

Authors and Affiliations

  • L. Rónyai
    • 1
    • 2
  • T. SzŐnyi
    • 2
  1. 1.Department of Computer ScienceThe University of ChicagoChicagoUSA
  2. 2.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations