# Chains on the Eggers tree and polar curves

### C.T.C. Wall

University of Liverpool, United Kingdom

## Abstract

If $B$ is a branch at $O∈C_{2}$ of a holomorphic curve, a Puiseux parametrisation $y=ψ(x)$ of $B$ determines "pro-branches" defined over a sector $∣argx−α∣<ε$ . 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 $C_{2}$, 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 $Γ$ at $O$ has contact with $C$ measured by a unique point $X_{Γ}∈T_{C}$, and this determines the set of exponents of contact with $C$ of any pro-branch of $Γ$. 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.

## Cite this article

C.T.C. Wall, Chains on the Eggers tree and polar curves. Rev. Mat. Iberoam. 19 (2003), no. 2, pp. 745–754

DOI 10.4171/RMI/367