Permutation polynomials in one and several variables

Various authors have dealt with problems relating to permutation polynomials over finite systems ([4], [8], [10], [18], [20]-[25],[29]-[33], etc.). In this thesis various known results are extended and several questions are resolved. Chapter 2 begins by considering the problem of finding those permutation polynomials in a single variable amongst some given classes of polynomials. Previously, this question was settled only for cyclic polynomials and Chebyshev polynomials of the first kind. Here we consider the Chebyshev polynomials of the second kind and polynomials of the form (x n- 1)/(x - I). Certain questions on multivariable polynomials are then considered. Chapter 3 deals with questions involving polynomials whose coefficients lie in a subfield of the given field, and considers some combinatorial questions. Chapter 4 resolves the structure of the group of maps of F nq ‚Äövúv¿ F nq induced by the extended Chebyshev polynomials of Lidl and Wells [26]. Chapter 5 extends this further to finite rings -Z/(pe), thus generalising results of Lausch-M‚â଱ller-N‚âàv´bauer [18]. Chapter 6 settles some questions concerning the conjecture of Schur on polynomials f(x) ˜ì¬µ Z[x] which permute infinitely many residue fields Fp. It is known ([10]) that these are compositions of cyclic and Chebyshev polynomials of the first kind. In chapter 6 it is determined which of these polynomials have the required property.