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On the difference between permutation polynomials. (English) Zbl 1374.11089
Summary: The well-known Chowla and Zassenhaus conjecture, proved by Cohen in 1990, states that if $$p>(d^2-3d+4)^2$$, then there is no complete mapping polynomial $$f$$ in $$\mathbb{F}_p[x]$$ of degree $$d\geq 2$$. For arbitrary finite fields $$\mathbb{F}_q$$, a similar non-existence result was obtained recently by Işık, Topuzoğlu and Winterhof in terms of the Carlitz rank of $$f$$.
Cohen, Mullen and Shiue generalized the Chowla-Zassenhaus-Cohen Theorem significantly in 1995, by considering differences of permutation polynomials. More precisely, they showed that if $$f$$ and $$f+g$$ are both permutation polynomials of degree $$d\geq 2$$ over $$\mathbb{F}_p$$, with $$p>(d^2-3d+4)^2$$, then the degree $$k$$ of $$g$$ satisfies $$k\geq 3d/5$$, unless $$g$$ is constant. In this article, assuming $$f$$ and $$f+g$$ are permutation polynomials in $$\mathbb{F}_q[x]$$, we give lower bounds for the Carlitz rank of $$f$$ in terms of $$q$$ and $$k$$. Our results generalize the above mentioned result of Işık et al. We also show for a special class of permutation polynomials $$f$$ of Carlitz rank $$n\geq 1$$ that if $$f+x^k$$ is a permutation over $$\mathbb{F}_q$$, with $$\operatorname{gcd}(k+1,q-1)=1$$, then $$k\geq(q-n)/(n+3)$$.

##### MSC:
 11T06 Polynomials over finite fields 14H05 Algebraic functions and function fields in algebraic geometry
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