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Variations on inversion theorems for Newton–Puiseux series

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Barroso, E. R. and González Pérez, Pedro Daniel and Popescu-Pampu, P (2017) Variations on inversion theorems for Newton–Puiseux series. Mathematische Annalen, 368 (3-4). pp. 1359-1397. ISSN 0025-5831

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Official URL: https://link.springer.com/article/10.1007/s00208-016-1503-1




Abstract

Let f (x, y) be an irreducible formal power series without constant term, over an algebraically closed field of characteristic zero. One may solve the equation f (x, y) = 0 by choosing either x or y as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and 1968 respectively, Abhyankar and Zariski published proofs of an inversion theorem, expressing the characteristic exponents of one set of series in terms of those of the other set. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the coefficients of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning irreducible series with an arbitrary number of variables.


Item Type:Article
Uncontrolled Keywords:Branch; Characteristic exponents; Plane curve singularities; Hypersurface singularities; Quasi-ordinary series; Lagrange inversion; Newton-Puiseux serie
Subjects:Sciences > Mathematics > Functions
Sciences > Mathematics > Algebraic geometry
ID Code:44737
Deposited On:22 Sep 2017 11:21
Last Modified:25 Sep 2017 08:18

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