Let p>2 be a prime. A function f: GF(p)→GF(p) is planar if for every a∃GF(p) * , the function f(x+a-f(x) is a permutation of GF(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. If P is a protective plane of prime order p admitting a collineation group of order p 2 , then P is the Galois plane PG(2, p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom  and our results are generalizations of some results of Johnson . We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.
- AMS subject classification (1980): 05B25, 11T21, 51E15
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics