Chains on the Eggers tree and polar curves

Abstract

If $B$ is a branch at $O\in\mathbb{C}^2$ of a holomorphic curve, a Puiseux parametrisation $y=\psi(x)$ of $B$ determines "pro-branches" defined over a sector $|\mathrm{arg} x-\alpha| < \varepsilon $. The exponent of contact of two pro-branches is the (fractional) exponent of the first power of $x$ where they differ. We first show how to use exponents of contact to give simple proofs of several well known results. For $C$ the germ at $O$ of a curve in $\mathbb{C}^2$, the Eggers tree $T_C$ of $C$ is defined. We also introduce combinatorial invariants (particularly, a certain 1-chain) on $T_C$. Any other germ $\Gamma$ at $O$ has contact with $C$ measured by a unique point $X_{\Gamma}\in T_C$, and this determines the set of exponents of contact with $C$ of any pro-branch of $\Gamma$. A simple formula establishes the converse, and this leads to a short proof of the theorem on decomposition of a transverse polar of $C$ into parts $P_i$, where both the multiplicity of $P_i$, and the order of contact with $C$ of each branch $Q$ of $P_i$ are explicitly given.