Variations on inversion theorems for Newton–Puiseux series

• Published in: Mathematische annalen Vol. 368; no. 3-4; pp. 1359 - 1397
• Main Authors: García Barroso, Evelia Rosa, González Pérez, Pedro Daniel, Popescu-Pampu, Patrick
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2017; Springer Nature B.V; Springer Verlag
• Subjects: Algebra; Exponents; Mathematical analysis; Mathematics; Mathematics and Statistics; Power series; Series (mathematics); Newton-Puiseux series; 32S25; Plane curve singularities; Characteristic exponents; Branch; Hypersurface singularities; 14H20; Primary 14B05; Quasi-ordinary series; Lagrange inversion; quasiordinary series; 2010 Mathematics Subject Classification. 14B05 (primary) 14H20 32S25 Branch Characteristic exponents plane curve singularities hypersurface singularities quasiordinary series Lagrange inversion Newton-Puiseux series; 2010 Mathematics Subject Classification. 14B05 (primary); plane curve singularities; hypersurface singularities; 32S25 Branch
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-016-1503-1

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.