Bases of subalgebras of K[[x]] and K[x]

Let $f\mathrm{\_}1,\dots ,f\mathrm{\_}s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[f1,\dots ,f\mathrm{\_}s]]\subseteq K[[x]]$ (respectively the semigroup of degrees of polynomials in$K[f\mathrm{\_}1,\dots ,f\mathrm{\_}s]\subseteq K[x]$). We give procedures to compute thesesemigroups and several applications.