Chains on the Eggers tree and polar curves

  • C.T.C. Wall

    University of Liverpool, United Kingdom


If BB is a branch at OC2O\in\mathbb{C}^2 of a holomorphic curve, a Puiseux parametrisation y=ψ(x)y=\psi(x) of BB determines "pro-branches" defined over a sector argxα<ε|\mathrm{arg} x-\alpha| < \varepsilon . The exponent of contact of two pro-branches is the (fractional) exponent of the first power of xx where they differ. We first show how to use exponents of contact to give simple proofs of several well known results. For CC the germ at OO of a curve in C2\mathbb{C}^2, the Eggers tree TCT_C of CC is defined. We also introduce combinatorial invariants (particularly, a certain 1-chain) on TCT_C. Any other germ Γ\Gamma at OO has contact with CC measured by a unique point XΓTCX_{\Gamma}\in T_C, and this determines the set of exponents of contact with CC of any pro-branch of Γ\Gamma. A simple formula establishes the converse, and this leads to a short proof of the theorem on decomposition of a transverse polar of CC into parts PiP_i, where both the multiplicity of PiP_i, and the order of contact with CC of each branch QQ of PiP_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