Permutation Polynomials Over Finite Fields: A Geometric Approach

The purpose of this paper is to utilize algebraic-geometric ideas in the study of polynomials for which the associated polynomial functions are permuta- tions of a given finite field. Polynomials of this type are called permutation polynomials. Due to the complexity of such geometric approach, we will concentrate on the specific study of low-degree polynomials over finite fields of characteristic 2. Nevertheless, the methods to be presented can be ap- plied to the study of polynomials of any degree over finite fields of arbitrary characteristic. Chapter 2 in this paper discusses criteria used in the determination of permutation polynomials, as well as elementary theoretical results. Other, more sophisticated results and some applications of permutation polyno- mials are discussed in Chapter 3. In Chapter 4, we relate concepts and ideas from both algebra and algebraic geometry to the study of low-degree polynomials over finite fields of characteristic 2; more specifically, we use algebraic geometry to classify permutation polynomials of degree up to 5 in characteristic 2. These same techniques are then applied, in Chapter 5, to the classification of general permutation polynomials (an important type of permutation polynomials) of degree 8 over finite fields of characteristic 2. To our knowledge, this is the first time this classification has been carried out.