Selected Title s i n Thi s Serie s 12 Salm a Kuhlmann , Ordere d exponentia l fields , 200 0 11 Tibo r Krisztin , Hans-Ott o Walther , an d Jianhon g Wu , Shape , smoothnes s an d invariant stratificatio n o f an attractin g se t fo r delaye d monoton e positiv e feedback , 199 9 10 Jif f Patera , Editor , Quasicrystal s an d discret e geometry , 199 8 9 Pau l Selick , Introductio n t o homotop y theory , 199 7 8 Terr y A * Loring , Liftin g solution s to perturbin g problem s i n C* -algebras, 199 7 7 S . O * Kochman, Bordism , stabl e homotop y an d Adam s spectra l sequences , 199 6 6 Kennet h R, . Davidson , C**Algebra s b y example, 199 6 5 A * Weiss, Multiplicativ e Galoi s module structure, 199 6 4 Gerar d Besson , Joachi m Lohkamp , Pierr e Pansu , an d Pete r Peterse n Miroslav Lovric , Maun g Min-Oo , an d McKenzi e Y.-K . Wang , Editors , Riemannian geometry , 199 6 3 Albrech t Bottcher , Aa d Dijksm a an d Hein z Langer , Michae l A . Dritsche l an d James Rovnyak , an d M . A . Kaashoe k Peter Lancaster , Editor , Lecture s o n operator theor y an d it s applications, 199 6 2 Victo r P a Snaith, Galoi s module structure , 199 4 1 Stephe n Wiggins , Globa l dynamics , phas e spac e transport, orbit s homoclini c t o resonances, an d applications , 199 3 This page intentionally left blank Ordered Exponential Fields This page intentionally left blank http://dx.doi.org/10.1090/fim/012 FIELDS INSTITUT E MONOGRAPHS THE FIELD S INSTITUT E FO R RESEARCH I N MATHEMATICA L SCIENCE S Ordered Exponential Fields Salma Kuhlman n American Mathematical Society Providence, Rhode Island The Field s Institut e for Researc h i n Mathematical Science s The Field s Institut e i s named i n honou r o f the Canadia n mathematicia n Joh n Charle s Fields (1863-1932) . Field s wa s a visionar y wh o receive d man y honour s fo r hi s scientifi c work, includin g election to the Royal Societ y of Canada i n 190 9 and to the Roya l Societ y o f London i n 1913 . Amon g othe r accomplishment s i n th e servic e o f the internationa l math - ematics community , Field s wa s responsibl e fo r establishin g th e world' s mos t prestigiou s prize fo r mathematic s research—th e Field s Medal . The Field s Institute fo r Researc h i n Mathematical Science s i s supported b y grants fro m the Ontario Ministr y o f Education an d Trainin g and th e Natural Science s and Engineerin g Research Counci l o f Canada . Th e Institut e i s sponsore d b y McMaste r University , th e University o f Toronto , th e Universit y o f Waterloo, an d Yor k Universit y an d ha s affiliate d universities i n Ontari o an d acros s Canada . This researc h wa s supporte d b y a Deutsch e Forschungsgemeinshaf t Habilitationssti - pendium an d a n Auslandsaufenthalts-Stipendium . Partiall y supporte d b y a n Individua l Research Gran t fro m th e Natura l Science s an d Engineerin g Researc h Counci l o f Canada , and b y th e Universit y o f Saskatchewa n President' s NSER C fund . 1991 Mathematics Subject Classification. Primar y 03C60 , 12J15 ; Secondary 12L12 , 26A12 . ABSTRACT. W e provide a detailed valuatio n theoreti c descriptio n o f ordered field s whic h admi t a n exponential function . I n particular , w e analyz e th e structur e o f the non-archimedea n model s o f o-minimal expansion s o f the reals , i n whic h th e exponentia l functio n i s definable . W e appl y ou r results to study the Hard y fields associate d to suc h expansions. Th e appendi x present s the mode l theory o f the valu e group s o f ordered exponentia l fields. Library o f Congres s Cataloging-in-Publicatio n Dat a Kuhlmann, Salma , 1958 - Ordered exponentia l fields / Salm a Kuhlmann . p. cm . — (Field s Institut e monographs , ISS N 1069-527 3 ; 12 ) Includes bibliographica l reference s an d index . ISBN 0-8218-0943- 1 (acid-fre e paper ) 1. Model theoreti c algebra . 2 . Ordered fields. I . Title. II . Series . QA9.7.K84 200 0 511'.8-dc21 99-04950 2 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission should be addressed to the Assistant to the Publisher, America n Mathematical Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionQams.org. © 200 0 by the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepared b y Th e Field s Institute . Visit th e AM S hom e pag e a t URL : http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 0 5 0 4 0 3 0 2 01 0 0 a me s soeurs Magda, Nawa l Farida et Amir a et me s filles Anna Nour a et Nail a This page intentionally left blank II faut fair e d e la vie un rev e et d u rev e une realit e Marie Curi e This page intentionally left blank Contents Introduction xii i Chapter 0 . Preliminarie s o n valued an d ordere d module s 1 1. Value d module s 1 2. Valuatio n independenc e 5 3. Ordere d module s 7 Chapter 1 . Non-archimedea n exponentia l fields 1 5 1. Th e natural valuatio n o f an ordered field 1 5 2. Th e skeleto n o f (K >0, • , 1, <) 1 8 3. Formall y exponentia l fields 2 2 4. Lexicographi c (de)compositio n o f exponentials 2 4 5. Exponentiatio n i n power serie s fields 2 7 6. Extension s an d maximalit y 2 9 7. Th e structure theor y fo r countabl e exponentia l fields 3 1 Chapter 2 . Valuatio n theoretic interpretation o f the growth and Taylor axioms 3 3 1. Th e axio m scheme s (GA ) an d (T ) 3 3 2. (GA)-exponential s an d the valu e group 3 4 3. Liftin g ex p from th e residue field 3 6 4. (T)-exponential s o n the infinitesimal s 3 7 5. Conclusio n 3 9 6. Countabl e exponentia l fields with growt h properties 4 0 7. Natura l contraction s arisin g fro m logarithm s 4 4 Chapter 3 . Th e exponentia l ran k 4 9 1. Conve x valuations 4 9 2. Th e exponential analogu e o f the rank 5 2 3. (GA) - and (Ti)-prelogarithm s 5 3 4. Th e shif t ma p & 5 6 5. Characterizatio n o f the exponential an d the principa l exponentia l ran k 6 1 Chapter 4 . Constructio n o f exponential fields 6 5 1. ^-Logarithmi c cross-section s 6 5 2. A combinatorial resul t an d it s consequences 6 7 3. Existenc e o f logarithmic cross-section s 7 0 4. Fro m prelogarithm s to logarithms 7 3 Chapter 5 . Model s fo r th e elementar y theor y o f th e real s wit h restricte d analytic function s an d exponentiatio n 7 7 1. Twistin g a group cross-sectio n b y an automorphism 7 7 Xll Contents 2. Th e exponential-logarithmi c powe r serie s field 7 9 3. Model s o f arbitrary principa l exponentia l ran k 8 3 Chapter 6 . Exponentia l Hard y fields 8 9 1. Som e basic valuation theor y 8 9 2.