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On the difference between permutation polynomials over finite fields
The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990,
states that if , then there is no complete mapping polynomial
in \Fp[x] of degree . For arbitrary finite fields \Fq, a
similar non-existence result is obtained recently by I\c s\i k, Topuzo\u glu
and Winterhof in terms of the Carlitz rank of .
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 and are both permutation
polynomials of degree over \Fp, with , then the
degree of satisfies , unless is constant. In this
article, assuming and are permutation polynomials in \Fq[x], we
give lower bounds for in terms of the Carlitz rank of
and . Our results generalize the above mentioned result of I\c s\i k et
al. We also show for a special class of polynomials of Carlitz rank that if is a permutation of \Fq, with , then
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