On the difference between permutation polynomials over finite fields

Abstract

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that if $p>(d^2-3d+4{)}^2$, then there is no complete mapping polynomial $f$ in $\Fp[x]$ of degree $d\ge 2$. For arbitrary finite fields $\Fq$, a similar non-existence result is obtained recently by I\c sık, Topuzo\u glu 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 $d\ge 2$ over $\Fp$, with $p>(d^2-3d+4{)}^2$, then the degree $k$ of $g$ satisfies $k\ge 3d/5$, unless $g$ is constant. In this article, assuming $f$ and $f+g$ are permutation polynomials in $\Fq[x]$, we give lower bounds for $k$ in terms of the Carlitz rank of $f$ and $q$. Our results generalize the above mentioned result of I\c sık et al. We also show for a special class of polynomials $f$ of Carlitz rank $n\ge 1$ that if $f+x^k$ is a permutation of $\Fq$, with $\gcd (k+1,q-1)=1$, then $k\ge (q-n)/(n+3)$.