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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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##### References:
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