Permutation polynomials and resolution of singularities over finite fields. (English) Zbl 0711.11050

Let \(F_ q\) denote the finite field of order q. In 1966 L. Carlitz conjectured that for each positive integer n, there is a constant \(C_ n\) such that for each finite field of odd order \(q>C_ n\), there does not exist a permutation polynomial of degree n over \(F_ q\). The conjecture is quite easily shown to be true for n a power of two but is only known to be true for the additional values \(n=6,10,12\), and 14. The author verifies the truth of the Carlitz conjecture for \(n=2\ell\) where \(\ell\) is an odd prime. This is accomplished by relating the conjecture to the study of the resolution of singularities of a plane algebraic curve over a finite field.
It should also be pointed out that, independently, S. D. Cohen has obtained the same result and has shown the conjecture to be true for all \(n<1000\) as well. Cohen’s method is based on exceptional polynomials over finite fields and on the theory of primitive permutation groups. His paper is to appear in Arch. Math. While both methods certainly have merit, it is not clear to the reviewer which method has the greater probability of leading to a solution of the entire conjecture. A complete proof of the conjecture would indeed be a major result.
Reviewer: G.L.Mullen


11T06 Polynomials over finite fields
11G20 Curves over finite and local fields
14H20 Singularities of curves, local rings
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