Good and bad field generators
Peter B. Russell · Kyoto journal of mathematics · 1977
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 [ I ] a n d [2].Using methods o f [2 ], 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 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 a, b E k [x , y ] such that (a, b)k [x , y] =k [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.