Short parametrization

Definition

The definition of short parametrization comes form the following result.

Proposition

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, \(S = \Gamma(f) = \langle r_0, r_1, \ldots, r_h \rangle\) the numerical semigroup associated to f with \(r_0 = n\) and \(r_1 = m\). Then, modulo an automorphism of \(\mathbb{K}[[x,y]]\), \(f\) has a parametrization of the form \(x = t^n\) and \(y = t^m + \sum_{q \in G(S), q > m} c_q t^q\), where \(G(S)\) denotes the set of gaps.

Let us consider the hypothesis of the proposition. It is said that a parametrization of \(f\) is a short parametrization of \(C(f)\) if it has an expression as above.

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.