Bases of subalgebras of K〚x〛 and K[x]

Let f1,…,fs be formal power series (respectively polynomials) in the variable x. We study the semigroup of orders of the formal series in the algebra K〚f1,…,fs〛⊆K〚x〛 (respectively the semigroup of degrees of polynomials in K[f1,…,fs]⊆K[x]). We give procedures to compute these semigroups and several applications. We prove in particular that the space curve parametrized by f1,…,fs has a flat deformation into a monomial curve.