\titleofpresentation{On the Cycle Structure of Permutation Polynomials of Finite Fields}

\presenter{Ay\c ca \c Ce\c smelio\u glu} 

This is joint work with Wilfried Meidl and Alev Topuzo\u glu.
Permutation polynomials over $\mathbb{F}_q$ of degree $\leq q-1$ form a group under composition and subsequent reduction modulo $x^q-x$ which is isomorphic to $S_q$.
Leonard Carlitz observed in his 1953 article \cite{Cesmelioglu1} that any transposition $(0\;a)$, $a \in \mathbb{F}_q$ can be represented by the polynomial 
\[P_a(x) = -a^2(((x-a)^{q-2} + a^{-1})^{q-2}-a)^{q-2}.\]
His result shows that the group of permutation polynomials over $\mathbb{F}_q$ is generated by $x^{q-2}$ and $ax+b$, $a,b \in \mathbb{F}_q$, $a \neq 0$. The above mentioned isomorphism motivates the study of the cycle structure of permutation polynomials of $\mathbb{F}_q$. See \cite{Cesmelioglu3}, \cite{Cesmelioglu4} for work on cycle structure of some specific polynomials of interest.

In this work we study the cycle structure of the polynomials
\[\mathcal{P}_n(x) = (((a_0x+a_1)^{q-2}+a_2)^{q-2}+ \ldots + a_n)^{q-2} + a_{n+1}\]
over $\mathbb{F}_q$ with $a_0 \neq 0$ and note that any permutation polynomial of $\mathbb{F}_q$ can be expressed in this form for some $n \geq 0$, with $\mathcal{P}_0(x) = a_0x+a_1$. 
We define a rational function corresponding to $\mathcal{P}_n(x)$ and by using the known structure of $\mathcal{P}_1(x)$ or the cycle structure of rational linear transformations we determine the cycle structure of $\mathcal{P}_2(x)$ and  $\mathcal{P}_3(x)$ completely . With the help of \cite{Cesmelioglu2} we also give constructions for $P_n(x)$ with full cycle for arbitrary $n \geq 1$.


\begin{ourbiblio}{99}  

\bibitem{Cesmelioglu1} 
L. Carlitz,  
Permutations in a finite field,
\emph{Proc. Amer. Math. Soc.} {\bf 4} (1953) 538. 

\bibitem{Cesmelioglu2} 
I. Beck, 
Cycle decomposition by transpositions.  
\emph{J. Combinatorial Theory Ser. A} {\bf 23} no. 2 (1977), 198--207. 

\bibitem{Cesmelioglu3} 
R. Lidl, G.L. Mullen, 
Cycle structure of Dickson permutation polynomials, 
\emph{Math. J. Okayama Univ.} {\bf 33} (1991) 1--11.

\bibitem{Cesmelioglu4} 
W.-S. Chou, I.E. Shparlinski, 
On the cycle structure of repeated exponentiation modulo a prime,
\emph{J. Number Theory}  {\bf 107} no. 2 (2004), 345--356.

\end{ourbiblio}

\presenteraddress{Sabanc{\i} University, Istanbul} 

\email{cesmelioglu@su.sabanciuniv.edu}
