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Schur covers and Carlitz’s conjecture. (English) Zbl 0855.11063
In this seminal paper the authors prove the Carlitz conjecture from 1966 which says that if $$q$$ is an odd prime power, then any exceptional polynomial over the finite field $$F_q$$ must have odd degree. Recall that a polynomial over $$F_q$$ is called exceptional if it is a permutation polynomial of infinitely many finite extensions of $$F_q$$. For the historical background on the Carlitz conjecture and equivalent formulations [see R. Lidl and G. L. Mullen, Am. Math. Mon. 95, 243-246 (1988; Zbl 0653.12010), ibid. 100, 71-74 (1993; Zbl 0777.11054)].
The starting point of the proof of the conjecture is what the authors call the exceptionality lemma, which gives a characterization of exceptional polynomials in terms of their geometric-arithmetic monodromy group pair. Both monodromy groups naturally come with permutation representations, and this establishes an important link with the theory of permutation groups. Another crucial ingredient of the proof is the ramification theory of covers of curves in positive characteristic.
The paper contains a wealth of additional information on exceptional polynomials. For example, it is shown that if $$F_q$$ has characteristic $$p>3$$, then any exceptional polynomial over $$F_q$$ is a composition of cyclic polynomials, Chebyshev polynomials, and polynomials of degree a power of $$p$$. This is a consequence of the result that for $$p>3$$ the arithmetic monodromy group of an indecomposable exceptional polynomial is an affine group, and the proof of this result relies in turn on the classification of finite simple groups.

##### MSC:
 11T06 Polynomials over finite fields 14H30 Coverings of curves, fundamental group 20D30 Series and lattices of subgroups 12E10 Special polynomials in general fields
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