A note on permutation polynomials and finite geometries

Let F be a finite field of odd cardinality. A polynomial g in F[x] is called a permutation polynomial if g defines a bijective function on F. We will call a polynomial f in F[x] a difference permutation polynomial if f(x + a) -f(x) is a permutation polynomial for every nonzero a in F. Difference permutation polynomials are a special case of planar functions, as defined by Dembowski [2, p. 2271, and give rise to affine planes. Quadratic polynomials are clearly difference permutation polynomials, and give rise to desarguesian planes. For F a prime field, J.F. Dillon (private communication) has asked whether there are any nonquadratic difference permutation polynomials over F. Such polynomials would give rise to nondesarguesian planes. An analogous question for arbitrary fields of odd cardinality is brought up in [3, p. 2571. In this note we show that every difference permutation polynomial over GF(p) is quadratic. The idea of the proof is roughly as follows. First we show that the graph of a difference permutation polynomial f is closely related to a (p + l)-arc K in the desarguesian projective plane over GF(p). Then Segre’s theorem says that K is a conic. This implies that f is a quadratic polynomial. I have been informed that the result of this paper has been independently obtained by Y. Hiramine. His proof is different from mine and was submitted for publication at about the same time.