Exceptional polynomials of affine type

[摘要] Let K be a finite field of characteristic p. A polynomial f with coefficients in K is said to be receptional if it induces a permutation on infinitely many finite extensions of K. Let t be a transcendental, and (K) over bar 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 (K) over bar 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 p(m) 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 p(2) will be. (C) 1997 Academic Press.

[发布日期] 1997-08-15