Canonical bases of modules over one dimensional k-algebras

Let K be a field and denote by K[t], the polynomial ring with coefficients in K. Set A = K[f1,. .. , fs], with f1,. .. , fs $\in$ K[t]. We give a procedure to calculate the monoid of degrees of the K algebra M = F1A + $\times$ $\times$ $\times$ + FrA with F1,. .. , Fr $\in$ K[t]. We show some applications to the problem of the classification of plane polynomial curves (that is, plane algebraic curves parametrized by polynomials) with respect to some oh their invariants, using the module of K{\"a}hler differentials

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

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.Comment: MEGA'2015 (Special Issue), Jun 2016, Trento, Ital