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 $\Q$-algebra $K$, and $\G'$ be the monoid of $K$-algebra monomorphisms of $K[x]$ of the type $\s : x\mapsto x+\l_2x^2+... +\l_nx^n$, $\l_i\in K$, $\l_n$ is a unit of $K$. It is proved that for each $\s \in \G'$ there are only finitely many distinct decompositions $\s = \s_1... \s_s$ in $\G'$. Moreover, each such a decomposition is uniquely determined by the degrees of components: if $\s = \s_1... \s_s= \tau_1... \tau_s$ then $\s_1=\tau_1, >..., \s_s=\tau_s$ iff $\deg (\s_1)=\deg (\tau_1), ..., \deg (\s_s)=\deg (\tau_s)$. Explicit formulae are given for the components $\s_i$ via the coefficients $\l_j$ and the degrees $\deg (\s_k)$ (as an application of the inversion formula for polynomial automorphisms in {\em several} variables from \cite{Bav-inform}). 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).