Exact degree of an algebroid curve

Definition

Let \(f(x,y) = y^n + a_1(x)y^{n-1} + \cdots + a_n(x) \in \mathbb{K}((x))[y]\) be an irreducible polynomial, and let \(x(t), y(t)\) be a short parametrization of \(f\). Let \(\mathbf{A} = \mathbb{K}[[x(t), y(t)]]\) and let \(\mathbf{M} = x'(t) \mathbf{A} + y'(t) \mathbf{A}\). It is defined

\[ I = \{ord_t(g) ~ | ~ g \in \mathbf{M} \}. \]

It can be proven that \(I\) is a relative ideal of \(S = \Gamma(f)\), the numerical semigroup associated to f. It is said that \(i \in I\) is an exact degree if \(i+1 \in S\). The other elements are said non exact degrees of \(\mathbf{M}\). The set of non exact degrees is denoted by \(NE(\textbf{M})\).

References

Assi, Abdallah, Marco D’Anna, and Pedro A. Garcı́a-Sánchez. 2020. Numerical Semigroups and Applications. Vol. 3. RSME Springer Series. Springer, [Cham]. https://doi.org/10.1007/978-3-030-54943-5.