Numerical semigroup associated to a curve

Definition

The definition of numerical semigroup associated to a curve needs the following proposition.

Proposition

Let \(F = y^n + a_1(x)y^{n-1} + \cdots + a_n(x)\) be a nonzero polynomial of \(\mathbb{K}[x][y]\) and assume, possibly after a change of variables, that \(deg_x ~ a_i(x) < i\) for all \(i \in \{1, 2, \ldots, n\}\) such that \(a_i(x) \ne 0\). Let \(h_F(u,x,y) = u^n F \left( \frac{x}{u}, \frac{y}{v} \right)\), \(F_{\infty}(u,y) = h_F(u,1,y)\) and \(f(x,y) = F(x^{-1}, y) \in \mathbb{K}[x^{-1}, y]\).

  1. \(f(x,y) = x^{-n}F_{\infty}(x, y)\).

  2. \(F\) has one place at infinity if, and only if, \(f(x,y)\) is irreducible in \(\mathbb{K}((x))[y]\).

Let us consider the hypothesis of the proposition and also that \(F\) has one place at the point at infinity. It is defined the numerical semigroup associated to \(f\) as

\[ \Gamma(f) = \{int(f,g) ~ | ~ g \in \mathbb{K}[x^{-1}][y] \setminus (f)\}. \]

It can be proven that \(\Gamma(f)\) is a subsemigroup of \(- \mathbb{N}\).

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.