  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 s\i k, 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 $k \geq 3d/5$, unless $g$ is constant. In this
article, assuming $f$ and $f+g$ are permutation polynomials in $\Fq[x]$, we
give lower bounds for $k %=\mathrm{deg(h)} $ in terms of the Carlitz rank of
$f$ and $q$. Our results generalize the above mentioned result of I\c s\i k et
al. We also show for a special class of polynomials $f$ of Carlitz rank $n \geq
1$ that if $f+x^k$ is a permutation of $\Fq$, with $\gcd(k+1, q-1)=1$, then
$k\geq (q-n)/(n+3)$.

Algebraic Geometry