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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 e 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 NewtonPuiseux expansion of a branch, is not available.

L e t k b e a f ie ld . A fie ld generator i n tw o variables o v e r k i s a polynomial f e k [x , y ] s u c h th a t k (x , y )= k (f, g ) f o r som e rational function g E k(x, y). W e continue the investigation of field generators begun i n, w e first study in detail properties o f th e multiplicity tree at infinity of f once coordinate functions x, y h a v e been chosen that are natural for f (see [2 , 4 .7 ]). O u r o rig in a l motivation f o r th is h a d b e e n a n attem pt to show th at all field generators are good in th e sense that a complementary genera to r g c a n b e fo u n d i n k [x , y ] . However, a quite astonishing exam ple o f a b a d field generator has been constructed by C . J a n i n [1 ], a n d w e instead use th e numerical information obtained to determ ine, w ith th e help of a com puter, a ll b a d field generators o f degree < 25, th e degree o f Jan's exam ple. W e find th a t field generators a r e g o o d f o r degrees d < 2 0 a n d d=22, 23, 24, a n d that there is exactly o n e " ty p e " o f b a d field generator f o r d = 2 1 a n d d = 25 (see 2.6 fo r a m o re precise statem ent). R . Ganong helped materially with th e rather elaborate calculations needed to establish this a n d w ith t h e w riting o f an appendix in which some o f th e details are explained. A g o o d fie ld generator f appears a s p a r t o f a birational morphism cp:AZ->AZ w it h tp(a, 13)=(f(a, fi), g(a, 13)) f o r a, 13 e k. W e s h o w th a t th is is a lm o st tru e i n g e n era l. N am e ly , if f i s a f ie ld generator, a complementary generator g =a lb can alw ays be found w ith x , y ] . This m eans that th e pencil o f curves { g-pLuEk} h a s no base points at finite distance a n d th a t 9: A t -4 P , yo(cc, fl)=(I, f(a, fl), g(a, 13)), i s a birationa l morphism.