Milnor number

Definition

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 and let \(f_x, f_y\) the partial derivatives of \(f\). It is defined the Milnor number, denoted by \(\mu(f)\), as \(\mu(f) = int(f_x, f_y)\), where \(int(f_x, f_y)\) denotes the intersection multiplicity of \(f_x\) and \(f_y\).

It can be proven that \(\int(f_x, f_y) = rank_{\mathbb{K}}(\mathbb{K}[[x]][y] / (f_x, f_y))\).

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.