Exceptional polynomials of affine type

Abstract

Let K be a finite field of characteristic p. A polynomial f with coefficients in K is said to be exceptional if it induces a permutation on infinitely many finite extensions of K. Let t be a transcendental, and K̄ be an algebraic closure of K. The exceptional polynomials known to date are constructed from certain twists of polynomials g, such that the splitting field of g(X) - t over K̄ is rational. On the other hand we know that except for p = 2 or 3, either the degree of an exceptional polynomial f is a power of the characteristic p, or f is a Dickson polynomial. The case deg f = p has been settled completely. We give a new series of exceptional polynomials of degree pm with m even which does not follow the above mentioned construction principle. We show under certain additional hypotheses that the associated monodromy groups of exceptional polynomials will be severely restricted. In particular, we determine precisely what the geometric monodromy groups of exceptional polynomials of degree p2 will be. © 1997 Academic Press.