Canonical bases of modules over one dimensional $\mathbf{K}$-algebras

Abstract

Let ${\mathbf {K}}$ be a field and denote by ${\mathbf {K}}[t]$, the polynomial ring with coefficients in ${\mathbf {K}}$. Set $\mathbf A={\mathbf {K}}[f_1,\ldots ,f_s]$, with $f_1,\ldots , f_s \in \mathbf K[t]$. We give a procedure to calculate the monoid of degrees of the ${\mathbf {K}}$ algebra ${\mathbf {M}}=F_1{\mathbf {A}}+\cdots +F_r{\mathbf {A}}$ with $F_1,\ldots , F_r\in {\mathbf {K}}[t]$. 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 of their invariants, using the module of Kähler differentials.