Definition
The definition of very short parametrization comes from the following result.
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 \(r_0 = n, r_1 = m\) and let \(G(S)\) be the set of gaps. Then, modulo an automorphism of \(\mathbb{K}[[x,y]]\), \(f\) has a parametrization of the form \(x = t^n\) and \(y = t^m + \sum_{q \in G(S), q > m} c_q t^q\), such that if \(y - t^m \ne 0\) and \(\lambda = \inf( Supp(y - t^m))\), where \(Supp(y-t^m)\) denotes the support of \(y\) and \(t^m\), then \(n + \lambda \not \in S\).
Let us consider the hypothesis of the proposition. It is said that a parametrization of \(f\) is a very short parametrization of \(f\) if it has an expression as above and satisfies the condition mentioned.
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.