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On permutation polynomials. (English) Zbl 1044.11103
The author studies the question when a polynomial of the form \(f(x)=x^u(x^v+1)\) with positive integers \(u,v\) induces a permutation on the finite field \(\mathbb F_q\). For \(d=3\) and \(d=5\) he gives sufficient and necessary conditions for \(f\) to be a permutation polynomial over \(\mathbb F_q\) where \(d\mid q-1\) and \(\gcd(v,q-1)=(q-1)/d\). The proof is based on Hermite’s criterion for permutation polynomials.
Remark: The numerous inductions in the proof of Lemma 4 can be evaded. Because of the symmetry of binomial coefficients, we have \(M(2n,3,c)=M(2n,3,2n-c)\) for all \(c\) and \(M(2n,3,c+1)=M(2n,3,c)+1\) whenever \(2n+c\equiv2\bmod 3\). With \(M(2n,3,0)+M(2n,3,1)+M(2n,3,2)=2^{2n}\), this yields Lemma 4.

MSC:
11T06 Polynomials over finite fields
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