Characterization of Polynomial Prime Bidecompositions

Bidecompositions, i.e., solutions to r o p = s o q, play a central role in the study of uniqueness properties of prime decompositions of polynomials with respect to functional composition. In [Ritt, 1922] all bidecompositions using polynomials over the complex number field have been characterized. Later the result was generalized to more general fields. All proofs tend to be rather long and involved. The object of this paper is to develop a version that is simpler than the existing ones, while keeping completely elementary. Thus it tries to give a better insight into this deep theorem, for a wider community.