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

Abstract: Let $f_1,\ldots, f_s$ be formal power series (respectively polynomials) in thevariable $x$. We study the semigroup of orders of the formal series inthe algebra $K[[ f1,\ldots, f_s]] \subseteq K[[ x ]]$ (respectively the semigroup of degrees of polynomials in$K[f_1,\ldots,f_s]\subseteq K[x]$). We give procedures to compute thesesemigroups and several applications.