Abstract
Let k be a field. To any pair (x,y) ≠ (0,0) of elements in tk[[t]], and hence to any irreducible and residually rational power series f ∈ k[[X,Y]], we associate a matrix, the Hamburger-Noether tableau of (x,y), which, in essence, is a description of the algorithm used to compute an element of minimal order in the integral closure of k[[x,y]] by successive quadratic transformations. Our main aim then is to reprove basic results of Abhyankar and Moh on approximate roots of polynomials and their use in the study of branches of plane curves, and to extend these results to certain situations over fields of positive characteristic where one important technical tool of Abhyankar and Moh, the Newton-Puiseux expansion of a branch, is not available.
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Russell, P. Hamburger-Noether expansions and approximate roots of polynomials. Manuscripta Math 31, 25–95 (1980). https://doi.org/10.1007/BF01303268
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DOI: https://doi.org/10.1007/BF01303268