FACTORIZATION OF MONOMORPHISMS OF A POLYNOMIAL ALGEBRA IN ONE VARIABLE

Abstract

Let K[x] be a polynomial algebra in a variable x over a commutative -algebra K, and Γ′ the monoid of K-algebra monomorphisms of K[x] of the type σ: x ↦ x+λ₂x² + . . . +λ_nxⁿ, λ_i ∈ K, λ_n is a unit of K. It is proved that for each σ ∈ Γ′ there are only finitely many distinct decompositions σ = σ₁. . .σ_s in Γ′. Moreover, each such decomposition is uniquely determined by the degrees of components: if σ = σ₁. . . σ_s= τ₁ . . . τ_s then σ₁=τ₁, λ. . ., σ_s=τ_s if and only if deg(σ₁)=deg(τ₁), . . ., deg(σ_s)=deg(τ_s). Explicit formulae are given for the components σ_i via the coefficients λ_j and the degrees deg(σ_k) (as an application of the inversion formula for polynomial automorphisms in several variables from [1]). In general, for a polynomial there are no formulae (in radicals) for its divisors (elementary Galois theory). Surprisingly, one can write such formulae where instead of the product of polynomials one considers their composition (as polynomial functions).