On polynomials over finite fields with particular value sets

Official URL: http://risc01.sabanciuniv.edu/record=b1649233 (Table of Contents)

A classical result on value sets of non-permutation polynomials over finite fields is due to Wan (1993). Denoting the cardinality of the value set of f 2 Fq[x] by jVf j, Wan's result gives the upper bound JVx, where d is the degree of f. A proof of this bound due to Turnwald, which was obtained by the use of symmetric polynomials is given in Chapter 2. A generalization of this result was obtained by Aitken that we also describe here. The work of Aitken focuses on value sets of pairs of polynomials in Fq[x], in particular, he studies the size of the intersection of their value sets. We present pairs of particular polynomials whose value sets do not only have the same size but are actually identical. Clearly, a permutation polynomial f of Fq[x] satisfies jVf j = q. In Chapter 3, we discuss permutation behaviour of pairs of polynomials in Fq[x].