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Generalized Newton-Puiseux Theory and Hensel's Lemma in C[[x, y]]

Published online by Cambridge University Press:  20 November 2018

Tzee-Char Kuo
Tzee-Char Kuo*
Affiliation:
University of Sydney, Sydney, Australia
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The Newton polygon and the Newton-Puiseux algorithm ([3], p. 370, [8], p. 98), and their generalizations, serve as a powerful tool for analysing the singularities of a given function. Yet experts know how difficult it is to keep track of them when one, or several, blowing-ups are applied. Thus many interesting theorems are stated under the strong, rather undesirable, assumption that the Newton faces are non-degenerate.

In this paper, we introduce a method which is parallel to the classical Newton-Puiseux theory, yet avoids blowing-ups and fractional power series, except in the proofs.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 41 , Issue 6 , 01 December 1989 , pp. 1101 - 1116
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Abhyankar, S.and Moh, T. T., Newton-Puiseux expansion and generalized Tshianhausen transformation II, J. reine und angew Math 261 (1973), 29–54.Google Scholar
2. Abhyankar, S., Irreducibility Criteria for germs of analytic functions of two complex variables, Advances in Math, to appear.Google Scholar
3. Brieskorn, E.and Knorrer, H., Plane algebraic curves (Birkhäuser, 1986).Google Scholar
4. Eisenbud, D.and Neumann, W., Three-dimensional link theory and invariants of plane curve singularities, Annals of Maths Studies 110 (Princeton University Press, 1985).Google Scholar
5. Kuo, T. C. and Lu, Y. C., On analytic function germs of two complex variables, Topology 16 (1977), 299–310.Google Scholar
6. Moh, T. T., On approximate roots of a polynomial, J. reine und angew. Math. 2781279 (1975), 301–306.Google Scholar
7. Oka, M., On the stability of the Newton boundary, Proceeding of Symposia in Pure Math 40, part 2 (1983), 259–268.Google Scholar
8. Walker, R., Algebraic curves (Princeton University Press, 1950).Google Scholar