If is a branch at of a holomorphic curve, a Puiseux parametrisation of determines "pro-branches" defined over a sector . The exponent of contact of two pro-branches is the (fractional) exponent of the first power of where they differ. We first show how to use exponents of contact to give simple proofs of several well known results. For the germ at of a curve in , the Eggers tree of is defined. We also introduce combinatorial invariants (particularly, a certain 1-chain) on . Any other germ at has contact with measured by a unique point , and this determines the set of exponents of contact with of any pro-branch of . A simple formula establishes the converse, and this leads to a short proof of the theorem on decomposition of a transverse polar of into parts , where both the multiplicity of , and the order of contact with of each branch of 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