INTRODUCTION

Polynomials are a classical subject of mathematics. The first steps

towards the abstract concept of a polynomial were the investigation of

algebraic equations and the theory of real and complex functions

the form

notions of a field and ring at the beginning of this century subsequently

brought about the development of the abstract concept of a polynomial

over a commutative ring with identity. While polynomials over the fields

of real or complex numbers play an important r61e in analysis and numerical

mathematics, the algebraic properties of polynomials have also

been a research object for a great number of papers and under various

different points of view.

There are, however, features in the algebraic theory of classical polynomials

that have been treated inseveral papers to some extent, but so

far have not been given a coherent representation. Among all these

aspects we think the most important ones to be the connection between

polynomials and polynomial functions and the properties of polynomial

functions. In particular, the so-called permutation polynomials over finite

fields (i.e. polynomials which represent permutations) have suggested

plenty of interesting algebraic and number-theoretical investigations.

Moreover we think that an important and interesting part of the theory

consists of the composition of polynomials. Questions concerning the

decomposition of polynomials into indecomposable factors, permutable

polynomials, or congruences which are compatible with the composition

operation, have been tackled by various authors.

It may appear somewhat strange that polynomials over commutative

rings with identity have been dealt with quite extensively while polynomials

Over other classes of algebraic structures, such as groups, semigroups,

lattices etc. have been given little attention. Those papers on polynomials

Over classes other than rings, fields, and maybe Boolean algebras, are

Scarce and scattered, and so far not even a general agreement on basic

definitions has been achieved. The first author who has endeavoured to

f off ( x ) = a,$+. . . +alx+ao. The introduction of the abstract1X