Exceptional Polynomials of Affine Type Original Research Article

LetKbe a finite field of characteristicp. A polynomialfwith coefficients inKis said to beexceptionalif it induces a permutation on infinitely many finite extensions ofK. Lettbe a transcendental, andimagebe an algebraic closure ofK. The exceptional polynomials known to date are constructed from certain twists of polynomialsg, such that the splitting field ofg(X) − toverimageis rational. On the other hand we know that except forp = 2 or 3, either the degree of an exceptional polynomialfis a power of the characteristicp, orfis a Dickson polynomial. The case deg f = phas been settled completely. We give a new series of exceptional polynomials of degreepmwithmeven 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 degreep2will be.