Open Access
September, 2003 Chains on the Eggers tree and polar curves
C. T. C. Wall
Rev. Mat. Iberoamericana 19(2): 745-754 (September, 2003).


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.


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C. T. C. Wall. "Chains on the Eggers tree and polar curves." Rev. Mat. Iberoamericana 19 (2) 745 - 754, September, 2003.


Published: September, 2003
First available in Project Euclid: 8 September 2003

zbMATH: 1057.14032
MathSciNet: MR2023205

Primary: 14H20

Keywords: Decomposition , Eggers tree , exponent of contact , polar curve

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.19 • No. 2 • September, 2003
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