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Further results on complete permutation monomials over finite fields. (English) Zbl 07067778
Summary: In this paper, we construct several new classes of complete permutation monomials $$a^{- 1} x^d$$ over a finite field $$\mathbb{F}_{q^n}$$ with exponents $$d = \frac{q^n - 1}{q - 1} + 1$$, $$\frac{q^{p - 1} - 1}{q - 1} + 1$$, and $$\frac{q^{q - 1} - 1}{q - 1} + 1$$, respectively, where $$q = p^k$$ is a power of a prime number $$p$$. Our approach uses the AGW criterion (the multiplicative case) together with Dickson permutation polynomials and a class of exceptional polynomials respectively. One of our results confirms Conjecture 4.18 by G. Wu, N. Li, T. Helleseth, Y. Zhang in [42] under the assumption that the characteristic $$p$$ is primitive modulo a prime number $$n + 1$$. Moreover, we show that Conjecture 4.18 is false in general using our approach and a counterexample is provided. We also re-confirm Conjecture 4.20 in [42] that was proved recently in [24], and extend some of these recent results to more general $$n$$’s and more general $$a$$’s.

##### MSC:
 11T06 Polynomials over finite fields 05A05 Permutations, words, matrices 11T55 Arithmetic theory of polynomial rings over finite fields
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