turkmath.org

The well-known Chowla and Zassenhaus conjecture, proven by Cohen in 1990, states that for any $d\ge 2$ and any prime $p>(d^2-3d+4)^2$ there is no complete mapping polynomial in $\mathbb{F}_p[x]$ of degree $d$. For arbitrary finite fields $\mathbb{F}_q$, we give a similar result in terms of the Carlitz rank of a permutation polynomial rather than its degree. We prove that if $n<\lfloor q/2\rfloor$, then there is no complete mapping in $\mathbb{F}_q [x]$ of Carlitz rank $n$ of small linearity. We also determine how far permutation polynomials $f$ of Carlitz rank $n<\lfloor q/2\rfloor$ are from being complete, by studying value sets of $f(x)+x$. We provide examples of complete mappings if $n=\lfloor q/2\rfloor$, which shows that the above bound cannot be improved in general. In this talk, we will also present a new method for constructing complete mappings of finite fields. We give a sufficient condition for the construction of a family of complete mappings of Carlitz rank at most $n$. Moreover, for $n=4,5,6$ we obtain an explicit construction of complete mappings. Finally, we discuss value sets of particular classes of polynomials over finite fields. We consider a class $\mathcal{F}_{q,n}$ of polynomials of the form $F(x)=f(x)+x$, where $f$ is a permutation polynomial of Carlitz rank at most $n$. The study of the spectrum of $\mathcal{F}_{q,n}$ enables us to obtain a simple description of polynomials $F \in \mathcal{F}_{q,n}$ with prescribed the value set $V_F$, especially those avoiding a given set, like cosets of subgroups of the multiplicative group $\mathbb{F}_q^*$. The value set count for such $F$ can also be determined. This yields polynomials with evenly distributed values, which have small maximum count.