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

A Recursive Construction of Permutation Polynomials over Fq2\mathbb{F}_{q^2} with Odd Characteristic from R\'{e}dei Functions

In this paper, we construct two classes of permutation polynomials over $\mathbb{F}_{q^2}$ with odd characteristic from rational R\'{e}dei functions. A complete characterization of their compositional inverses is also given. These permutation polynomials can be generated recursively. As a consequence, we can generate recursively permutation polynomials with arbitrary number of terms. More importantly, the conditions of these polynomials being permutations are very easy to characterize. For wide applications in practice, several classes of permutation binomials and trinomials are given. With the help of a computer, we find that the number of permutation polynomials of these types is very large

Inversion Polynomials for Permutations Avoiding Consecutive Patterns

In 2012, Sagan and Savage introduced the notion of $st$-Wilf equivalence for a statistic $st$ and for sets of permutations that avoid particular permutation patterns which can be extended to generalized permutation patterns. In this paper we consider $inv$-Wilf equivalence on sets of two or more consecutive permutation patterns. We say that two sets of generalized permutation patterns $\Pi$ and $\Pi'$ are $inv$-Wilf equivalent if the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $\Pi$ is equal to the generating function for the inversion statistic on the permutations that simultaneously avoid all elements of $\Pi'$. In 2013, Cameron and Killpatrick gave the inversion generating function for Fibonacci tableaux which are in one-to-one correspondence with the set of permutations that simultaneously avoid the consecutive patterns $321$ and $312.$ In this paper, we use the language of Fibonacci tableaux to study the inversion generating functions for permutations that avoid $\Pi$ where $\Pi$ is a set of five or fewer consecutive permutation patterns. In addition, we introduce the more general notion of a strip tableaux which are a useful combinatorial object for studying consecutive pattern avoidance. We go on to give the inversion generating functions for all but one of the cases where $\Pi$ is a subset of three consecutive permutation patterns and we give several results for $\Pi$ a subset of two consecutive permutation patterns