The Milnor number of plane irreducible singularities in positive characteristic (Q2788657)

Let \(\mathbb K\) be an algebraically closed field of characteristic \(p>0\), \(f\) an irreducible power series in \(\mathbb K[[x,y]]\) and \(\Gamma (f)\) the semigroup associated with \(f\). Let \(\bar{\beta}_0,\dots,\bar{\beta}_g\) be the minimal sequence of generators of \(\Gamma (f)\). The authors relate two important invariants of the irreducible singularity \(f\): the Milnor number NEWLINE\(\mu (f):=\dim_\mathbb K\mathbb K[[x,y]]/(\frac{\partial f}{\partial x},\frac{\partial f}{\partial y})\) and the degree of the conductor \( c(f):=\) the smallest element of \(\Gamma (f)\) such that \(c(f), c(f)+1,\dots \in \Gamma (f)\). If \(p>\text{ord}f\) then \(\mu (f)=c(f)\) if and only if \(\bar{\beta}_k\neq 0\) (mod \(p\)) for \(k=1,\dots,g\).