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Exceptional Polynomials of Affine Type

✍ Scribed by Robert M. Guralnick; Peter Müller

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
319 KB
Volume
194
Category
Article
ISSN
0021-8693

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✦ Synopsis

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 K be an algebraic closure of K. The exceptional polynomials known to date are constructed from certain twists of Ž . polynomials g, such that the splitting field of g X y t over K is rational. On the other hand we know that except for p s 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 s 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.

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