Contact

Definition

The definition of contact needs the following result.

Proposition

Let \(f(x,y) = y^n + a_1(x)y^{n-1} + \cdots + a_n(x) \in \mathbb{K}((x))[y]\). Suppose that \(f(x,y)\) is irreducible. Then, there exists \(y(t) \in \mathbb{K}((t))\) such that \(f(t^n, y(t)) = 0\) and there are finite polynomials satisfying this property.

Let \(f \in \mathbb{K}((x))[y]\) be a monic irreducible polynomial of degree \(n\) in \(y\) and let \(y_1(t), \ldots, y_q(t)\) be the roots of \(f(t^n, y) = 0\). Let \(g \in \mathbb{K}((x))[y]\) be another monic irreducible polynomial of degree \(p\) in \(y\), and let \(z_1(t), \ldots, z_p(t)\) be the set of roots of \(g(t^p, y) = 0\). It is defined the contact of \(f\) with \(g\), denoted by \(c(f,g)\), as

\[ c(f,g) = \frac{1}{np} \max_{i,j} ord_t (y_i(t^p) - z_j(t^n)). \]

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.