Milnor number of plane curve singularities in arbitrary characteristic

TL;DR

This work extends Kouchnirenko-type formulas to plane curve singularities in arbitrary characteristic by developing the Hamburger-Noether algorithm and Newton trees. It computes the invariant $ar{ ext{μ} }(f)=2\, ext{δ}(f)-r(f)+1$ from areas under a sequence of Newton polygons associated to HN-transforms, without degeneration hypotheses, and relates this to the Milnor number via precise divisibility conditions on the vertices’ multiplicities $N_v$. The authors prove $2\δ(f)=-M(\mathcal{T})+r(f)$ and show $ ext{μ}(f)\ge \bar{\text{μ}}(f)$ in general, with equality in characteristic zero and in several positive-characteristic cases (non-degenerate, large $p$, and under assumptions aligning with existing conjectures). They also connect the Newton-tree formalism to the Zariski semigroup of irreducible branches, intersection multiplicities, and delta-invariants, providing a computational framework for singularities in arbitrary characteristic and unifying prior results by Greuel–Nguyen and García-Barroso–Płoski.

Abstract

Reduced power series in two variables with coefficients in a field of characteristic zero satisfy a well-known formula that relates a codimension related to the normalization of a ring and the jacobian ideal. In the general case Deligne proved that this formula is only an inequality; García Barroso and Płoski stated a conjecture for irreducible power series. In this work we generalize Kouchnirenko's formula for any degenerated power series and also generalize García Barroso and Płoski's conjecture. We prove the conjecture in some cases using in particular Greuel and Nguyen.

Figures (11)

• Figure 1: Left figure corresponds to the part of the Newton tree associated to the edge $v_i$. The central one is the Newton tree of the right figure.
• Figure 2: New Newton tree and gluing.
• Figure 3: Newton tree of Example \ref{['ex1']}.
• Figure 4: Newton polygon of Example \ref{['ex2']}.
• Figure 5: Newton tree of Example \ref{['ex2']} if $\mathop{\mathrm{Char}}\nolimits\mathbb{K}\neq 2$ (to the left), or $\mathop{\mathrm{Char}}\nolimits\mathbb{K}= 2$ (to the right).

Theorems & Definitions (53)

• Conjecture
• Lemma 1.1
• proof
• Lemma 1.2
• proof
• Lemma 1.3
• Lemma 1.4
• proof
• Definition 1.5
• Remark 2.1