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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 \(k\geq(q-n)/(n+3)\).
11T06 Polynomials over finite fields
14H05 Algebraic functions and function fields in algebraic geometry
Full Text: DOI
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