On complete mappings and value sets of polynomials over finite fields

In this thesis we study several aspects of permutation polynomials over nite elds with odd characteristic. We present methods of construction of families of complete mapping polynomials; an important subclass of permutations. Our work on value sets of non-permutation polynomials focus on the structure of the spectrum of a particular class of polynomials. Our main tool is a recent classi cation of permutation polynomials of Fq, based on their Carlitz rank. After introducing the notation and terminology we use, we give basic properties of permutation polynomials, complete mappings and value sets of polynomials in Chapter 1. We present our results on complete mappings in Fq[x] in Chapter 2. Our main result in Section 2.2 shows that when q > 2n + 1, there is no complete mapping polynomial of Carlitz rank n, whose poles are all in Fq. We note the similarity of this result to the well-known Chowla-Zassenhaus conjecture (1968), proven by Cohen (1990), which is on the non-existence of complete mappings in Fp[x] of degree d, when p is a prime and is su ciently large with respect to d. In Section 2.3 we give a su cient 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. Chapter 3 is on the spectrum of the class Fq,n of polynomials of the form F(x) = f(x)+x, where f is a permutation polynomial of Carlitz rank at most n. Upper bounds for the cardinality of value sets of non-permutation polynomials of the xed degree d or xed index l were obtained previously, which depend on d or l respectively. We show, for instance, that the upper bound in the case of a subclass of Fq,n is q -2, i.e., is independent of n. We end this work by giving examples of complete mappings, obtained by our methods

On the difference between permutation polynomials over finite fields

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 $k$ 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 $k\geq (q-n)/(n+3)$

Can a Drinfeld module be modular?

Let $k$ be a global function field with field of constants \Fr and let $\infty$ be a fixed place of $k$. In his habilitation thesis \cite{boc2}, Gebhard B\"ockle attaches abelian Galois representations to characteristic $p$ valued cusp eigenforms and double cusp eigenforms \cite{go1} such that Hecke eigenvalues correspond to the image of Frobenius elements. In the case where k=\Fr(T) and $\infty$ corresponds to the pole of $T$, it then becomes reasonable to ask whether rank 1 Drinfeld modules over $k$ are themselves ``modular'' in that their Galois representations arise from a cusp or double cusp form. This paper gives an introduction to \cite{boc2} with an emphasis on modularity and closes with some specific questions raised by B\"ockle's work.Comment: Final corrected versio

Permutation polynomials and systems of permutation polynomials in several variables over finite rings

This paper will present the historical development of theorems regarding permutation polynomials in several variables over finite fields. Single variable permutation polynomials will be discussed since they are so important to the discussions which will follow. Theorems involving permutation polynomials and systems of permutation polynomials will also be considered. It will be shown that many of the interesting results obtained for finite fields can be generalized to finite rings

A Riemann Hypothesis for characteristic p L-functions

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute values. Further we use absolute values to give similar reformulations of the classical conjectures (with, perhaps, finitely many exceptional zeroes). We show how both sets of conjectures behave in remarkably similar ways.Comment: This is the final version (with new title) as it will appear in the Journal of Number Theor