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Approximate roots. (English) Zbl 1036.13017
Kuhlmann, Franz-Viktor (ed.) et al., Valuation theory and its applications. Volume II. Proceedings of the international conference and workshop, University of Saskatchewan, Saskatoon, Canada, July 28–August 11, 1999. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3206-9/hbk). Fields Inst. Commun. 33, 285-321 (2003).
From the paper: Given an integral domain $$A$$, a monic polynomial $$P$$ of degree $$n$$ with coefficients in $$A$$ and a divisor $$p$$ of $$n$$, invertible in $$A$$, there is a unique monic polynomial $$Q$$ such that the degree of $$P-Q^p$$ is minimal for varying $$Q$$. This $$Q$$, whose $$p$$-th power best approximates $$P$$, is called the $$p$$-th approximate root of $$P$$. If $$f\in\mathbb{C} [[X]][Y]$$ is irreducible, there is a sequence of characteristic approximate roots of $$f$$, whose orders are given by the singularity structure of $$f$$. This sequence gives important information about this singularity structure. We study its properties in this spirit and we show that most of them hold for the more general concept of semiroot. We show then how this local study adapts to give a proof of Abbyankar-Moh’s embedding line theorem:
If $$\mathbb{C} [X,Y]\to \mathbb{C} [T]$$ is an epimorphism of $$\mathbb{C}$$-algebras, then there exists an isomorphism of $$\mathbb{C}$$-algebras $$\mathbb{C}[U,V] \to \mathbb{C}[X,Y]$$ such that the composed epimorphism $$\mathbb{C} [U,V] \to \mathbb{C}[T]$$ is given by $$U=T$$, $$V=0$$.
For the entire collection see [Zbl 1021.00011].

##### MSC:
 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 14B05 Singularities in algebraic geometry 14R15 Jacobian problem 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems) 32B15 Analytic subsets of affine space 13F25 Formal power series rings