Intersection multiplicity

Definition

The definition of intersection multiplicity comes from 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 \(f(t^n, y(t)) = 0\).

Let \(f\) as in the proposition, \(g \in \mathbb{K}((x))[y]\) with \(g \ne 0\) and \(y(t)\) a root of \(f(t^n, y(t)) = 0\). It is defined the intersection multiplicity of \(f\) with \(g\), denoted by \(int(f,g)\), as

\[ int(f,g) = ord_t g(t^n, y(t)) \]

It can be proven that this definition does not depend on the root \(y(t)\), that is, \(int(f,g) = ord_t g(t^n, y(wt))\) for all \(w \in \mathbb{K}\) such that \(w^n = 1\).

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.