Numerical semigroup associated to a polynomial

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. It is defined the numerical semigroup associated to the polynomial \(f\), denoted by \(\Gamma(f)\), as

\[ \Gamma(f) = \{int(f,g) ~ | ~ g \in \mathbb{K}[[x]][y] \setminus (f) \}, \]

where \(int(f,g)\) denotes the intersection multiplicity of \(f\) with \(g\).

It can be proven that for any polynomial \(f\) monic and irreducible in \(\mathbb{K}[[x]][y]\), \(\Gamma(f)\) is a free numerical semigroup generated by \(\underline{r} = (r_0, r_1, \ldots, r_h)\), where \(\underline{r}\) comes from the definition of characteristic sequences of \(f\).

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.