Characterization of Polynomial Prime Bidecompositions a Simplified Proof

Bidecompositions, i.e., solutions to r ◦ p = s ◦ q, play a central rôle in the study of uniqueness properties of complete decompositions with respect to functional composition. In [Rit22] 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 making it accessible to a wider community. 1. Bidecompositions In the whole paper we will deal with polynomials over a field, which is usually denoted by k. The indeterminant, or identity polynomial, will be denoted by x. Thus, whenever we just say polynomial, elements of k[x] are intended, if not specified differently. Their functional composition will be denoted by ◦. Thus for polynomials r and p we use the notations r ◦ p = r ◦ p(x) = r(p(x)) = r(p) interchangably. The degree of a polynomial p is denoted by [p]. As usual when dealing with composition, we use the convention [0] = 0. Thus the degree function is a homomorphism of the monoid (k[x], ◦) onto the monoid (N0, ·). The units of (k[x], ◦) are the polynomials of degree 1. We have to clarify some basic notions. 1.1. Definition. 1. Two polynomials p and q are called associated, in symbols p ∼ q, whenever there exist units a and b such that p = a ◦ q ◦ b. 2. A polynomial f is said to be decomposable iff it has a decomposition f = r ◦ p into two non-units r and p. Otherwise it is called 1