Free semigroups of power series

Abstract

Given power series $f_1,\dots ,f_r\in z^{2}K[[z]]$ over a field $K$, we show that the semigroup $⟨f_1,\dots ,f_r⟩$ generated by the $f$'s under composition is free of rank $r$ whenever the multiplicative semigroup $⟨\mathrm{lc}(f_1),\dots ,\mathrm{lc}(f_r)⟩$ in $K^{\times }$ is free commutative of rank $r$; here $\mathrm{lc}(f)$ denotes the first nonzero coefficient of $f$. To do this, we associate an affine linear transformation to every $F\in ⟨f_1,\dots ,f_r⟩$ and then apply a ping-pong lemma. In particular, combining this with work of Pappalardi, Sha, Shparlinski, and Stewart, it follows that most semigroups of polynomials over a number field $K$ are free, and we make this statement precise through a counting argument.