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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
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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. Hard y fields 9 2 3. Valu e groups 9 7 4. Th e Hard y field o f a polynomially bounde d + (exp ) expansio n 9 8 5. Exponentia l boundednes s 10 1 6. Level s 10 2 7. Th e Crucia l Lemma fo r model s o f Tan 10 3 8. Residu e fields o f .F-exp-log-closures 10 7 9. A truncation fre e solutio n to the Hardy proble m 11 1 10. Undefinabilit y o f the Rieman n ^-functio n 11 3 Appendix A . Th e mode l theory o f contraction group s 11 7 1. Preliminarie s 11 7 2. Cut s i n ordered Abelia n group s 11 8 3. Ordere d abelia n group s with contraction s 12 0 4. Wea k o-minimalit y 13 6 Bibliography 15 5
Index 15 9 List o f Notatio n Introduction
The aim of this monograph is to describe the models of the elementary theory of an o-minimal expansio n o f the real s i n which the exponential functio n i s definable . We focus on polynomially bounded + (exp ) expansions (cf . Sectio n 4 of Chapter 6). Important example s o f suc h expansion s are : th e expansio n b y restricte d analyti c functions and the unrestricted exponential function (cf . [D—M—Ml]) , the expansion by convergen t generalize d powe r serie s an d th e unrestricte d exponentia l functio n (in which the Riemann ^-function restricte d to (1 , oo) is definable: cf . [D—SI]) , and the expansion by multisummable real power series and the unrestricted exponentia l function (i n which the T-functio n restricte d t o (0 , oo) i s definable: cf . [D—S2]) . The notion o f o-minimality (cf . Sectio n 2 of Chapter 6 ) was introduced b y van den Drie s i n [D3] , whil e studyin g th e expansio n (R,exp ) o f th e ordere d field o f the rea l number s b y th e rea l exponentia l function . Va n de n Drie s observe d tha t the subset s o f the cartesia n product s R n whic h ar e parametricall y definabl e i n a n o-minimal expansio n o f R share many o f the geometric properties o f semi-algebrai c sets. Fo r example , a semi-algebrai c se t ha s onl y finitely man y connecte d com - ponents, eac h o f them semi-algebrai c (cf . [CO]) . Va n de n Drie s showe d tha t thi s result remain s true i f one replaces "semi-algebraic " b y "parametricall y definabl e i n an o-minimal expansio n o f R" (cf . als o cell-decomposition fo r o-minima l structure s [K-P-S]). Thi s i s a finiteness theorem, an d va n den Drie s has se t out a s a goa l to explain the other finiteness phenomen a i n real algebraic and real analytic geometr y as consequence s o f o-minimalit y (cf . [D4]) . Th e breakthroug h wa s achieve d wit h Wilkie's results on the o-minimality o f the reals with exponentiation (cf . [Wl]) . H e showed that the expansion o f R by Pfaffian function s restricted to the closed unit box (i.e., the function s ar e set to be identicall y zer o outside the uni t box ) ha s a mode l complete theory . Thi s resul t ma y b e viewe d a s a strong refinemen t o f Gabrielov' s Theorem (cf . [GA]) . Th e latte r state s that th e clas s o f sub-analytic set s i s close d under takin g complements . Wilkie' s Theorem show s that i f the restricted analyti c functions use d to describe a given sub-analytic se t A ar e Pfaffian, the n the comple - ment o f A ma y als o be described b y Pfaffia n functions . Wilki e als o establishes th e model completeness o f the elementary theory T(exp) o f R with the real exponentia l function exp . Thi s theorem ha s a n importan t geometri c interpretatio n (cf . [Wl] , p. 1054) : cal l a subset o f Rn semi-exponential-algebraic (semi-EA ) i f it i s defined b y exponential-polynomial equations and inequalities, and a map from Rn t o Rm semi - EA i f its graph i s so, and finally a set to be sub-EA i f it i s the imag e o f a semi-E A set under a semi-EA map. The n the theorem i s equivalent to the assertion that th e complement o f a sub-EA se t i s a sub-EA set . This , a s fo r th e semi-algebrai c case , implies that th e clas s o f sub-EA set s i s also close d under takin g closures , interior s and boundaries . Recentl y (cf . [W2] ) Wilki e prove d a far-reachin g generalization , from whic h i t follow s tha t th e expansio n o f the real s b y total Pfaffia n function s i s o-minimal a s well.
xiii XIV Introduction
When proving results such as model completeness, the model theorist has to in- vestigate the class o f all models o f a given theory, instead o f studying one particular model. A t the center o f the class o f models that w e consider lie s the ordered field of the reals. Bu t i n this class there are also non-archimedean models , which we under- take to describe. Ou r basi c tool fo r this i s valuation theory . Inspire d b y the theor y of real place s an d rea l close d fields, an d seekin g the analog y t o the semi-algebrai c case, w e systematically develo p a n exponentia l analogu e fo r al l important notion s and methods . W e us e this abstrac t machiner y t o describ e explicitl y th e algebrai c structure o f th e models , an d t o giv e concret e constructions . Thes e construction s use powe r serie s fields (cf . Theore m 5. 7 and Theore m 5.11) . The valuatio n theor y tha t w e need i s basic. Indeed , w e consider ordere d fields with conve x valuations, whose residue fields are ordered an d hence o f characteristi c zero. Ther e are no deep mysteries in the theory o f valued fields in the characteristi c zero case . W e use , s o t o say , "shadows " o f valuations . I n fact , w e reduc e th e valuation theory to the "bones " b y going down to the skeletons o f the value groups, and ye t furthe r dow n to the value sets o f the valu e groups. Thi s descen t make s u s deal wit h thos e valu e sets , whic h ar e ofte n lexicographi c ordering s an d hav e som e kind o f ultrametri c structur e tha t i s reminiscen t o f th e valuation s wa y u p o n th e fields. At first sight , thi s approac h ma y see m to o simplistic . However , th e powe r o f this metho d come s fro m th e fac t tha t i n mos t importan t cases , a s i n th e cas e o f the powe r serie s fields, w e can lif t th e whol e situatio n fro m th e valu e se t bac k u p to the field, via the valu e group. At th e beginnin g o f thi s work , ou r approac h t o th e exponentia l functio n i s equally naive . Althoug h ou r ultimat e ai m i s to construc t extremel y well-behave d exponentials (tha t is , satisfyin g al l th e elementar y theorem s tha t th e rea l expo - nential functio n does) , w e start b y workin g o n ver y simpl e exponentials . W e just demand that the y are order preserving group isomorphisms fro m the additive grou p of the field ont o th e multiplicativ e grou p o f its positiv e elements . Bu t eve n thos e almost ridiculousl y weak exponentials impose tremendous restrictions on the struc- ture o f th e ordere d fields tha t carr y them . A t th e poin t o f thi s wor k wher e w e discover tha t powe r serie s fields cannot carr y exponentials , w e are force d t o mak e our demand s o n th e exponentials , (o r mor e precisel y o n thei r compositiona l in - verses, th e logarithms) eve n mor e modes t tha n before . Then , w e have to lear n t o work wit h prelogarithms , tha t i s non-surjectiv e logarithms . W e just hav e thes e logarithms tha t ar e no t eve n surjectiv e an d tha t enjo y n o reasonabl e propertie s whatsoever. Luckily however , w e hav e thre e importan t key s t o modif y an d improv e thos e weak prelogarithms, al l the wa y through . The first i s the discover y that powe r serie s fields, whilst no t carryin g surjectiv e logarithms, alway s carry prelogarithms. Thu s w e develop a standard metho d to get a surjectiv e logarith m o n a countable unio n o f power serie s fields. The secon d ke y i s a resul t o f [D—M—Ml ] wh o sho w tha t powe r serie s fields can be naturally made into models o f restricted rea l exponentiation, an d even more, of all restricted analyti c functions . Thi s structure i s preserved b y taking countabl e unions o f powe r serie s fields. Thu s w e no w en d u p wit h ordere d fields, endowe d with a n exponentia l whos e restriction to the unit box of the field enjoy s th e sam e elementary properties a s the real exponential restricted to the unit bo x in the reals. But w e ar e no t done . I n fac t th e exponentia l thu s constructe d doe s not yet satisfy th e elementary propertie s o f the unrestricted rea l exponential! Here , we use Introduction xv the las t ke y tha t open s th e las t door : Ressayre' s Theore m [RE] . I t state s tha t an exponential o n an ordered field which satisfie s th e elementar y propertie s o f th e restricted rea l exponential , and satisfie s th e growt h axio m schem e (GA) , satisfie s the elementary propertie s o f the unrestricted rea l exponential a s well. This leads us to a substantial part o f the research presented i n this monograph . We undertake a systemati c stud y o f the growt h axio m scheme . Ou r mai n ide a i s to investigate what (GA ) impose s on the value group o f the field. Descendin g eve n further down , we are able to encode the growth axiom scheme in the value set of the value group o f the field. Heuristically , (GA ) i s encoded in the order automorphism s which th e valu e se t carries . I t i s certainl y muc h easie r t o understan d a totall y ordered set (th e valu e set ) endowe d wit h a n automorphism , tha n t o understan d a totall y ordere d field endowe d wit h a logarithm . S o t o say , w e ge t ri d o f th e algebraic structure , bu t w e lose nothing. Indeed , a s mentione d already , w e ar e i n the privileged situation where we can lif t thi s information u p again, from the valu e set t o th e ordere d field. W e develo p a canonica l metho d t o achiev e thi s lifting . Finally, this allows us to describe the non-archimedean ordere d fields endowed wit h exponentials an d restricte d analyti c function s whic h satisf y al l the axiom s o f rea l exponentiation, an d whic h moreove r enjo y furthe r interestin g properties . Related to (GA ) i s the notion o f exponential rank. I t i s a finer measurement o f the growth rates of exponentials than just (GA) . We encode the exponential rank as well in the automorphisms o f the valu e set. W e construct fields o f arbitrary expo - nential ranks, the "exponential-logarithmi c powe r serie s fields". Thes e canonicall y defined exponentia l fields are the exponential analogu e o f power serie s fields in th e real case. The y can carry a multitude o f exponentials o f distinct growt h rates, an d enjoy furthe r surprisin g propertie s o f which w e provide a detailed account . Hardy fields provide the most beautifu l exampl e o f non-archimedean exponen - tial fields. The y wer e introduce d b y Hard y (cf . [HD2]) , a s "th e natura l domai n for th e stud y o f asymptoti c analysis" . W e appl y ou r genera l structur e theorem s for exponentia l fields to this particularly importan t case . Inspire d b y Rosenlicht' s work on Hardy fields o f finite rank (cf . [ROl]) , w e extend the study to the infinit e rank case . W e give a detailed descriptio n o f the valu e group s an d residu e fields o f Exponential Hard y fields (whic h necessarily have infinit e rank) . Usin g our results , we present at the end o f the monograph a new proof to a conjecture raised by Hardy concerning the asymptotic behaviour o f the "Logarithmico-Exponential " functions . This conjectur e wa s first establishe d i n [D— M—M2] by differen t methods . In Chapter 0 , we gather some preliminaries about valuations on ordered Abelian groups. W e introduce the valu e set an d the skeleto n o f such a group. Thes e invari - ants turn ou t to be very handy throughout thi s research . In Chapte r 1 , w e study th e necessar y condition s tha t a n ordere d exponentia l field K ha s to satisfy. W e show at the end of Chapter 2 that the exponential induce s canonically a n amazin g map (whic h w e call a contraction) o n the value group G of K wit h respect to the natural valuation. A contraction contracts every archimedean class 7 ^ {0} to a se t {a , —a} of two points, an d ye t map s G surjectively ont o G i n an order preserving way. Th e class of ordered Abelian groups that ar e able to carr y contractions i s much smalle r than the clas s o f all ordered Abelia n groups . I t i s a n elementarily axiomatizabl e clas s an d ha s a ver y well-behave d mode l theory . Thi s has been worked out i n [KF1 ] and [KF2] , an d w e present th e result s o f these tw o papers i n the Appendi x o f this monograph . At the end o f Chapter 1 , we study "small " non-archimedea n exponential fields. We investigate the following problem: Give n an ordered field K, an d assuming tha t xvi Introductio n
its residue field K wit h respect t o the natural valuation (whic h i s a subfield o f R) i s an exponentially close d subfield o f the reals, is it possible to lift the real exponentia l exp to an exponential / o f Kl W e answer this problem fo r non-archimedean count - able fields that ar e roo t close d fo r positiv e elements . W e ge t a structur e theore m for those fields (cf . Theore m 1.44) , and sho w that the y admit exponential s / liftin g exp fro m thei r residu e fields i f an d onl y i f thei r valu e grou p i s isomorphi c t o th e lexicographic su m o f copie s o f the additiv e ordere d grou p (if , +,0, <), take n ove r the rationals . I n Sectio n 6 o f Chapte r 2 , w e sho w tha t thi s conditio n i s indee d sufficient t o ge t exponential s satisfyin g (GA ) o n th e countabl e field. Thi s theo - rem provides a method t o construct non-archimedea n countabl e exponential fields, given archimedea n one s (cf . Exampl e 1.45) . In Chapte r 2 , we translate th e meanin g o f (GA ) an d the Taylo r axio m schem e (T) into a valuation theoretic language. Fo r example, we show that (T ) i s equivalent n to assertions of the form v(f(x)—E n(x)) > v(x ) (wher e En denote s the n-th partial sum i n th e Taylo r expansio n o f exp). Thes e result s ar e use d throughou t th e late r chapters. In Chapter 3 , we introduce prelogarithms and define the exponential rank to be the chain of convex valuation rings which are compatible with the prelogarithm. W e characterize th e exponentia l ran k throug h exponential equivalence (cf . Sectio n 4 of Chapter 3), as the rank is characterized b y the "multiplicativ e equivalence relation" in th e rea l case . I t i s worthwhil e mentionin g her e tha t i f K i s a mode l o f th e elementary theor y T o f a n exponentiall y bounde d expansio n o f th e reals , suc h that th e exponentia l / i s definable , the n th e valuatio n ring s R w o f valuation s w compatible wit h / ar e precisely th e T-conve x valuatio n ring s o f K , i n the sens e o f [D-L]. So far , w e hav e onl y describe d result s tha t ar e i n nic e analog y t o th e theor y of rea l places . Bu t whe n i t come s t o existenc e results , th e analog y break s down . If a field ha s a plac e ont o a n ordere d residu e field, the n th e orde r ca n b e lifte d up to th e field through th e place . I t i s not surprisin g tha t exponential s canno t b e lifted throug h arbitrar y places . Indeed , w e sho w i n Chapte r 4 that powe r serie s fields never admi t exponential s compatibl e wit h thei r canonica l valuation . (I t i s interesting to note that ther e i s an exponential o n the surreal numbers, cf . [G] , but this "powe r serie s field" i s a prope r class. ) However , w e sho w tha t ever y powe r series field R((G)) carrie s a prelogarithm . Indeed , i n Sectio n 3 o f Chapte r 4 , w e give an explici t formul a fo r th e basi c prelogarithm log 0 o n powe r serie s fields with any give n valu e grou p o f th e for m R r°, wher e T Q i s a totall y ordere d set . Goin g to th e unio n ove r a n increasin g chai n o f powe r serie s fields, w e mak e thi s loga - rithm log o surjective (cf . Sectio n 4 of Chapter 4) . W e call the so-obtained field the exponential-logarithmic power series field an d denot e i t b y R{(TQ)) EL. Th e loga - n rithm log 0 doe s °t satisf y (GA) , and w e develop a method i n Chapter 5 to modif y log0. W e show that on e can us e an y orde r preservin g map a o n T o satisfying tha t
Analogous constructions , usin g powe r serie s fields, ar e give n i n [D—M— M2]. There, a first limi t proces s i s employed to obtain a field with non-surjectiv e expo - nential, and then a second (inverse ) limit process renders the exponential surjective . The outcome i s a model called the "logarithmic-exponentia l powe r serie s field". I n contrast t o the constructio n give n i n [D—M— M2], our constructio n use s onl y on e limit process . I t i s an interestin g tas k fo r futur e researc h t o compar e th e model s obtained b y these tw o different approaches . Chapter 6 answer s a questio n raise d b y Macintyr e i n a cours e give n a t Th e Fields Institute , durin g th e Algebrai c Mode l Theor y Program , Novembe r 1996 . In [D—M— M2], the author s us e result s o f Ressayr e an d Mourgue s t o sho w tha t Hardy's field LE o f Logarithmico-exponential function s admit s a truncation-close d embedding in the logarithmic-exponential power series field (see above). Then , they use this particular embedding to prove Hardy's conjecture (cf . Section 9 of Chapter 6 for details ) an d to sho w that certai n functions , includin g the Gamma-functio n an d the Rieman n ^-function , canno t b e define d usin g exponentia l function , logarith m and restricte d analyti c functions . Whil e lecturin g o n th e result s o f [D—M-M2] , Macintyre aske d whethe r thei r result s coul d b e deduce d b y a "mor e invariant " version o f truncation. Indeed , w e derive the result s o f [D—M— M2] without usin g embeddings in the logarithmic exponential power series field. We replace truncation results by an intrinsic property, which is an assertion about the residue fields o f the Hardy fields with respec t t o arbitrary conve x valuations. I t i s invariant becaus e i t does no t depen d o n a n embeddin g i n logarithmic-exponentia l powe r serie s fields. As a by-product , w e get a structur e theore m fo r th e Hard y fields associate d t o a polynomially bounded + (exp ) expansion o f the reals (cf . Theore m 6.30 ) an d show , amongst othe r results, that thes e Hardy fields have levels in the sense o f Rosenlicht (cf. [ROSJ) . Several results presente d i n this monograp h wer e obtained i n the joint paper s [K-Kl], [K-K2] , [K-K3] , [K-K4 ] an d [K-K-Sl] . I woul d lik e t o than k m y co-authors Franz-Vikto r Kuhlman n an d Saharo n Shela h fo r thei r essentia l contri - butions i n ou r join t work . Specia l thank s ar e du e t o Franz-Vikto r Kuhlman n fo r allowing th e inclusio n o f result s o f [KF'l ] an d [KF2 ] a s a n Appendix , an d fo r proof-reading m y manuscript . I a m particularl y gratefu l t o Alexande r Preste l an d Pete r Roquett e fo r thei r constant support during the years I spent in Germany. I am grateful to Sudesh Kaur Khanduja fo r invitin g m e to teach a course o n exponential fields a t he r universit y (Chandigarh, India , 1995) , and to Charle s Delzel l an d Jame s Madde n fo r invitin g me t o giv e a mini-cours e o n thi s materia l durin g th e Specia l Semeste r o n Rea l Algebraic Geometr y an d Ordere d Structure s (Bato n Rouge , 1996) . I would like to thank Lo u van den Dries, Askold Khovanskii, Angus Macintyre , David Marker, John Shackell , Patrick Speissegge r an d Mark Spivakovsk y fo r usefu l conversations. I am especially endebted to Chris Miller , fro m who m I have learne d a lo t i n Baton Roug e and a t Th e Field s Institute . I am endebted t o John Marti n and the Mathematics Departmen t i n Saskatoo n for th e privileg e o f teachin g a n advance d graduat e cours e o n thi s materia l durin g the first year of my appointment. I thank the colleagues who attended my course fo r their constructiv e comments . Specia l thanks to Murra y Marshal l fo r alway s bein g demanding o n rigour an d clarity , an d fo r proof-readin g part s o f my manuscript . I acknowledg e th e suppor t fro m th e Deutsch e Forschungsgemeinschaft , bot h through a researc h gran t i n German y an d a specia l gran t t o atten d th e Progra m Year o n Algebrai c Mode l Theor y a t Th e Field s Institut e (1996-1997) . I than k xviii Introductio n
Br add Har t an d Mat t Valeriot e fo r organizin g thi s excitin g year , an d Th e Field s Institute fo r it s hospitality . Last bu t no t least , I wis h t o than k my husban d an d my childre n fo r thei r patience, and fo r thei r lovin g care. Salma Kuhlman n Saskatoon, Jun e This page intentionally left blank Bibliography
[A] Ailing , N . L.: On exponentially closed fields, Proc . Amer . Math . Soc . 13 (1962) , 706-71 1
[A-K] Ailing , N. L.- Kuhlmann, S.: On r]a-groups and fields, Order 11 (1994), 85-92 [BO] Boshernitzan , M. : Hardy fields and existence of trans exponential func- tions, Aequatione s Mathematica e 3 0 (1986) , 258-28 0 [BOU] Bourbaki , N.: Commutative algebra, Paris (1972 ) [BR] Brown , R. : Valued vector spaces of countable dimension, Publ . Math . Debrecen 1 8 (1971) , 149-15 1 [C] Cantor , G. : Beitrage zur Begrundung der transfiniten Mengenlehre, Math. Ann . 46 (1895) , 481-512 [CO] Collins , G . E.: Quantifier Elimination for Real closed Fields by Cylin- drical Algebraic Decomposition, in : Automat a Theor y and Forma l Lan - guage, 2n d G . I . Conferenc e Kaiserslautern , Springer , Berli n (1975) , 134-183 [C-K] Chang , C . C . - Keisler , H . J. : Model Theory, Amsterda m - Londo n (1973) [DA-G] Dahn , B. I. - Goring , P.: Notes on exponential logarithmic terms, Fund . Math. 12 7 (1986) , 45-50 [DA-WO] Dahn , B . I. - Wolter , H.: On the theory of exponential fields, Zeitschr . f. math . Logi k und Grundlage n d . Math . 2 9 (1983) , 465-48 0 [DL-WD] Dales , H . G . - Woodin , W . H. : Super-Real Fields; Totally Ordered Fields with Additional Structure, Londo n Math. Soc . Monographs, Ne w Series 14 , Clarendon Press , Oxfor d (1996 ) [D] va n de n Dries , L.: A generalization of the Tarski-Seidenberg theorem, and some nondefinability results, Bull . Amer . Math . Soc . 1 5 (1986) , 189-193 [Dl] va n de n Dries , L.: Tame topology and o-minimal structures, LM S Lec - ture Note s Serie s 248, Cambridg e Universit y Pres s (1998 ) [D2] va n den Dries, L.: T-convexity and tame extensions II, J . Symb . Logi c 62 (1997) , 14-3 4 [D3] va n de n Dries , L. : Remarks on Tarski's problem concerning (R, +,-,exp), in : Logi c Colloquiu m '82 , eds . G . Lolli , G . Long o an d A. Marcja, North-Hollan d (1984) , 97-12 1 [D4] va n de n Dries , L.: o-minimal structures, in : Logic : Fro m Foundation s to Applications . Europea n Logi c Colloquium , eds . W. Hodges , M . Hy - land, C . Steinhorn an d J . Truss , Oxfor d (1996) , 137-18 5
155 156 Bibliograph y
[D-L] va n den Dries, L. - Lewenberg , A. H.: T-convexity and tame extensions, J. Symb . Logi c 60 (1995) , 74-10 2 [D-M-Ml] va n den Dries, L. - Macintyre , A . - Marker , D.: The elementary theory of restricted analytic functions with exponentiation, Annal s Math . 14 0 (1994), 183-20 5 [D-M-M2] va n den Dries , L. - Macintyre , A . - Marker , D.: Logarithmic-Exponen- tial Power Series, t o appea r i n J. Londo n Math . Soc . [D-Sl] va n den Dries , L. - Speissegger , P. : The real field with convergent gen- eralized power series is model complete and o-minimal, t o appea r i n Trans. Amer . Math . Soc . [D-S2] va n de n Dries , L . - Speissegger , P. : The field of reals with multisum- mable series and the exponential function, preprin t [F] Fleischer , I. : Maximality and ultracompleteness in normed modules, Proc. Amer . Math . Soc . 9 (1958) , 151-15 7 [FU] Fnchs , L.: Partially ordered algebraic systems, Pergamo n Press, Oxfor d (1963) [G] Gonshor , H. : An introduction to the theory of surreal numbers, Cam - bridge Universit y Pres s (1986 ) [GA] Gabrielov , A. : Projections of semi-analytic sets, Functiona l Analysi s and it s Applications, 2 (1968) , 282-29 1 [G-J] Gillman , L.-Jerison , M . : Rings of continous functions, va n Nostrand , Princeton (1960 ) [GRAl] Gravett , K . A . H.: Valued linear spaces, Quart. J . Math . Oxfor d (2) , 6 (1955), 309-31 5 [GRA2] Gravett , K . A. H.: Ordered Abelian groups, Quart. J. Math. Oxfor d (2) , 7 (1956) , 57-6 3 [H] Hahn , H. : Uber die nichtarchimedischen Grofiensysteme, S.-B . Akad . Wiss. Wien, math.-naturw . Kl . Abt. Ila, 11 6 (1907) , 601-65 5 [HD1] Hardy , G . H.: Orders of Infinity, Cambridg e Universit y Pres s (1910 ) [HD2] Hardy , G . H.: Properties of Logarithmico-Exponential functions, Proc . London Math . Soc . (2 ) 1 0 (1912) , 54-9 0 [KA] Kaplansky , I. : Maximal fields with valuations I, Duk e Math . Journ . 9 (1942), 303-32 1 [KN-WR] Knebusch , M . - Wright , M. : Bewertungen mit reeller Henselisierung, J. rein e angew . Math. 286/287 (1976) , 314-32 1 [KF1] Kuhlmann , F.-V. : Abelian groups with contractions I, in : Abelia n Group Theory and Related Topics, Proceedings of the Oberwolfach Con - ference on Abelian Groups 1993 , (eds. R. Gobel , P. Hill and W. Liebert ) Amer. Math . Soc . Contemporary Mathematic s 17 1 (1994 ) [KF2] Kuhlmann , F.-V. : Abelian groups with contractions II: weak o-mini- mality, in : Abelia n Group s an d Modules . Proceeding s o f th e Padov a Conference 1994 , eds . A . Facchin i an d C . Menini , Kluwe r Academi c Publishers, Dordrech t (1995 ) [KF3] Kuhlmann , F.-V. : Valuation theory of fields, abelian groups and mod- ules, t o appea r i n th e "Algebra , Logi c an d Applications " series , eds . A. Macintyr e an d R . Gobel , Gordo n & Breach, Ne w York [KS1] Kuhlmann , S. : On the structure of nonarchimedean exponential fields I, Archiv e fo r Math . Logi c 34 (1995) , 145-18 2 Bibliography 157
[KS2] Kuhlmann , S. : Isomorphisms of Lexicographic Powers of the Reals, Proc. Amer . Math . Soc . 123 (1995) , 2657-266 2 [KS3] Kuhlmann , S. : Valuation bases for extensions of valued vector spaces, Forum Math . 8 (1996) , 723-73 5 [KS4] Kuhlmann , S.: Infinitary Properties of valued and ordered vector spaces, J. Symb . Logi c 64 (1999) , 216-22 6 [K-Kl] Kuhlmann , F.-V . - Kuhlmann , S. : On the structure of nonarchimedean exponential fields II, Comm . Alg . 22 (1994) , 5079-510 3 [K-K2] Kuhlmann , F.-V . - Kuhlmann , S. : The exponential rank, Th e Field s Institute Preprin t Serie s (1997) , to appea r in : Proc . o f the Specia l Se - mester o n Rea l Algebrai c Geometr y an d Ordere d Structures , Amer . Math. Soc . Contemporary Mathematic s Serie s [K-K3] Kuhlmann , F.-V . - Kuhlmann , S. : Residue fields of arbitrary convex valuations on restricted analytic fields with exponentiation I, The Field s Institute Preprin t Serie s (1996 ) [K-K4] Kuhlmann , F.-V . - Kuhlmann , S. : Explicit construction of exponential- logarithmic power series, preprin t in : Structure s Algebrique s Ordon - nees, Seminair e Pari s VII (1997 ) [K-K-Sl] Kuhlmann , F.-V . - Kuhlmann , S . - Shelah , S. : Exponentiation in Power Series Fields, Proc. Amer. Math. Soc . 12 5 (1997) , 3177-318 3 [K-K-S2] Kuhlmann , F.-V . - Kuhlmann , S . - Shelah , S. : Functional Equations in Lexicographic Products, preprin t [K-P-S] Knight , J . - Pillay , A . - Steinhorn , C. : Definable sets in ordered struc- tures II, Trans . Amer. Math. Soc . 295 (1986) , 593-60 5 [LAM1] Lam , T. Y.: The theory of ordered fields, in: Rin g Theory an d Algebr a III (ed . B . McDonald) , Lectur e Note s i n Pur e an d Applie d Math . 5 5 Dekker, Ne w York (1980) , 1-15 2 [LAM2] Lam , T . Y. : Orderings, valuations and quadratic forms, Amer . Math . Soc. Regiona l Conferenc e Serie s in Math. 52 , Providence (1983 ) [LAN] Lang , S.: The theory of real places, Ann. Math. 5 7 (1953) , 378-39 1 [LAU1] Laugwitz , D.: Eine nichtarchimedische Erweiterung angeordneter Kor- per, Math. Nachr . 37 (1968) , 225-23 6 [LAU2] Laugwitz , D.: Tullio Levi-Civita's work on nonarchimedean structures, in: Tulli o Levi-Civita , Convegn o internazional e celebrativ o de l cente - nario dell a nascita , Academi a Nazional e de i Lincei , Att i de i Convegn i Lincei 8 (1973 ) [LC] Levi-Civita , T.: Sugli infiniti ed infinitesimi attuali quali elementi ana- litici (1892-1893), Oper e mathematiche, vol . 1 , Bologna (1954) , 1-3 9 [MI] Miller , C. : Exponentiation is hard to avoid, Proc. Amer . Math . Soc . 122 (1994) , 257-25 9 [M-MI] Marker , D . - Miller , C. : Levelled o-minimal structures, Proc . o f th e Conference o n Rea l Algebrai c an d Analyti c Geometry , Revist a Mate - matica, Servici o D e Publicaciones Universida d Complutens e (1997 ) [MC-W] Macintyre , A.J.-Wilkie, A.J. : On the decidability of the real exponential field, Kreisel 70t h Birthday Volume , ed. P. G . Odifreddi , CLS I (1995 ) [MP-M-S] Macpherson , D. - Marker , D. - Steinhorn , C.: Weakly O-minimal struc- tures, preprint (1998 ) 158 Bibliograph y
[N] Neumann , B . H.: On ordered division rings, Trans . Amer . Math . Soc . 66 (1949) , 202-25 2 [P-S] Pillay , A . - Steinhorn , C. : Definable sets in ordered structures I, Trans. Amer. Math . Soc . 295 (1986) , 565-59 2 [PC] PrieB-Crampe , S. : Angeordnete Strukturen. Gruppen, Korper, projek- tive Ebenen, Ergebniss e de r Mathemati k un d ihre r Grenzgebiet e 98 , Springer (1983 ) [PO] Poizat , B.: Cours de theorie des modeles, Nu r e l Mantiq wa l Ma'arifa , Villeurbanne, (1985 ) [PR1] Prestel , A. : Lectures on Formally Real Fields, Springe r Lectur e Note s in Math. 1093 , Springer , Berlin-Heidelberg-Ne w York-Toky o (1984 ) [PR2] Prestel , A.: Einfuhrung in die mathematische Logik und Modelltheorie, Vieweg studium, Braunschwei g (1986 ) [RE] Ressayre , J.-P. : Integer parts of real closed exponential fields, in : eds . P. Clote and J . Krajicek , Arithmetic , Proo f Theor y an d Computationa l Complexity, Oxfor d Universit y Press , Ne w York (1993 ) [RI] Ribenboim , P. : Theorie des valuations, Le s Presse s d e l'Universit e d e Montreal, Montreal , 1s t ed . (1964) , 2n d ed . (1968 ) [RO] Robinson , A. : Functio n theor y o n som e nonarchimedea n fields , Amer . Math. Monthl y 8 0 (1973) , 87-10 9 [ROl] Rosenlicht , M. : Hardy fields, J . o f mathematical analysi s an d applica - tions 9 3 (1983) , 297-31 1 [R02] Rosenlicht , M. : The rank of a Hardy field, Trans . Amer . Math . Soc . 280 (1983) , 659-67 1 [R03] Rosenlicht , M. : Growth properties of functions in Hardy field, Trans . Amer. Math . Soc . 299 (1987) , 261-27 1 [R04] Rosenlicht , M. : On Liouville's theory of elementary functions, Pacifi c J. Math . 6 5 (1976) , 485-49 2 [S] Shackell , J.: Inverses of Hardy's L-functions, Bull . London Math . Soc . 25 (1993) , 150-15 6 [Wl] Wilkie , A . J.: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J . Amer . Math . Soc . 9 (1996) , 1051-109 4 [W2] Wilkie , A . J. : A general theorem of the complement and some new o-minimal structures, submitte d (1997 ) [Zl] Zil'ber , B.: Generalized analytic sets, Algebra i Logika, 36 (1997) , 361- 380 (i n Russian ) [Z2] Zil'ber , B. : Quasi-Riemann surfaces, in : Logic : fro m Foundation s t o Applications, Europea n Logi c Coll. , ed . W . Hodges , M . Hyland , C . Steinhorn an d J . Truss , Clarendo n Press , Oxfor d (1996) , 515-53 6 Index
Additive Lexicographic Decompositio n The - Cauchy sequence , 9 1 orem, 1 8 centrifugal, 46 , 12 1 algebraic, 13 6 centripetal, 46 , 12 1 X-algebraic, 13 1 chain, 1 algebraic closure , 13 6 characteristic domain , 138 , 13 9 algebraic exchang e property , 13 6 characteristic sequence , 131 , 13 9 Ailing, N. , 27 , 2 9 characterization archimedean of the exponentia l rank , 6 1 i?-archimedean, 9 of the principa l exponentia l rank , 6 3 equivalence class , 1 6 class archimedean componen t ^-equivalence class , 9 corresponding t o 7, 1 0 archimedean, 1 6 archimedean equivalent , 1 5 exponential, 5 8 asymptotic, 9 6 multiplicative, 5 2 automorphism closed increasing, 7 8 X£-closed, 6 1 axioms ^-closed, 6 1 n-th Taylo r axiom , 3 3 (^-closed, 6 1 (CO), (C<), (C-) , 12 0 ^"-closed, 9 7 (CA), (CS), (CP), (CF), (CSN), (CZ), (CC), rJT-closed, 9 7 (CA'), 12 1 exponentially closed , 3 2 (CE), 5 2 relatively exp-close d in , 10 8 (CO), 4 9 coarser, 5 0 (CV1), (CV2) , 12 1 coefficient ma p correspondin g t o 7 , 2 (CV3), (CV4) , (CV5) , 12 2 compatible, 2 3 (EH1), (EH2) , (EH3) , 9 4 incompatible exponential , 2 3 (GA), 3 3 v-compatible exponentia l field, 2 3 (NV1), (NV2) , (NV3) , (NV4) , 11 7 ^-compatible logarithm , 2 3 (NV5), 11 8 w-compatible exponential , 5 2 (OAG), 12 0 incompatible prelogarithm , 5 3 (Ti), 5 3 compatible wit h th e order , 9 , 4 9 (T), 3 3 complement, 1 0 (VO), (VI) , (V2) , (V3) , (V4) , 11 7 canonical additiv e complement , 2 7 growth axiom , 3 3 canonical multiplicativ e complement , 2 7 Taylor axiom , 3 3 component bounded corresponding t o 7 , 2 X-transcendental, 13 3 of x i n M, 8 exponentially, 9 5 constant term , 2 7 polynomially, 9 5 contraction, 44 , 12 1 Brown's Theorem , 6 centrifugal, 4 6 Brown, Ron , 6 centripetal, 4 6 natural, 4 5 canonical additiv e complement , 2 7 natural contractio n induce d b y h, 4 5 canonical grou p cross-section , 7 0 natural contractio n induce d b y /, 4 5 canonical multiplicativ e complement , 2 7 contraction group , 12 1 canonical valuation , 2 7 contraction hull , 12 5
159 160 Index
divisible, 12 5 (Tn)-exponential, 3 3 convex, 8 (T)-v-right exponential , 3 4 subgroup, 5 0 (T)-exponential, 3 3 convex partitio n group exponentia l induce d b y x> 45 finite, 13 8 induced o n the residu e field , 5 2 finite open , 14 0 lexicographic product , 2 5 convex subgrou p strong grou p exponential , 3 5 associated t o w, 5 0 exponential class , 5 8 convex subrin g exponential field, 2 2 principal, 5 1 v-compatible, 2 3 convex valuation , 16 , 4 9 extension o f exponential fields , 2 9
associated t o G w, 5 0 formally, 2 2 Countable Cas e Characterizatio n Theorem , exponential grou p i n A , 2 6 31 exponential Hard y field , 9 4 cross-section exponential rank , 5 2 ^-logarithmic, 6 6 characterization, 6 1 canonical grou p cross-section , 7 0 of a prelogarithm, 5 3 group cross-section , 7 0 principal, 6 1 of a valued field , 6 6 exponential-logarithmic powe r serie s fiel d strong v-logarithmic , 7 7 over To , 7 5
surjective w- logarithmic, 6 6 over (r 0,cr), 8 1 Crucial Lemma , 10 4 exponentially bounded , 9 5 cut, 11 8 exponentially closed , 3 2 i>G-cut, 11 8 exponents induced b y a n element , 11 8 field of , 9 7 shifted vc-cut , 11 9 extension immediate extensio n o f valued modules , 3 definable, 9 5 of exponential fields, 2 9 dense of valued modules , 3 u>-dense, 2 9 value se t preserving , 12 4 order dense , 3 0 divisible, 1 6 field o f exponents, 9 7 contraction hull , 12 5 final segmen t principal, 5 1 elements finer, 5 0 finite, 1 7 finite conve x partition, 13 8 infinitesimal, 1 7 finite elements , 1 7 positive infinite , 1 7 finite ope n conve x partition , 14 0 embedding formally exponentia l field, 2 2 truncation closed , 8 7 embedding o f valued modules , 2 germ a t oo , 9 2 equivalent Gravett, K . A . H. , 1 2 ^-equivalent t o y , 9 group (^-equivalent, 5 7 exponential grou p i n A, 2 6 equivalent valuation s , 2 of 1-units , 16 , 1 8 exchange propert y of positive units, 1 8 algebraic, 13 6 of units, 1 6 exponential, 2 2 strong exponential , 3 5 T\-exponential, 5 3 group cross-section , 7 0 ^-compatible, 2 3 canonical, 7 0 i>-left exponential , 2 4 group exponential , 2 6 ^-middle exponential , 2 5 induced b y x > 45 f-right exponential , 2 5 growth axiom , 3 3 w-compatible, 5 2 (GA)-i;-left exponential , 3 4 Holder's Theorem , 1 5 (GA)-exponential, 3 3 Holder, O. , 1 5
(GATn)-exponential, 3 3 Hahn Embeddin g Theorem , 1 4 (GAT)-i>-middle exponential , 3 4 Hahn product , 3 (GAT)-exponential, 3 3 Hahn sum , 3
(Tn)-v-right exponential , 4 2 Hardy field, 9 2 Index 161
exponential, 9 4 natural valuatio n levelled, 9 7 .R-natural valuation , 9 henselian, 2 8 on a n ordered field, 1 6 homomorphism o f ordered modules , 8 o-minimal, 9 5 immediate extensio n o-minimal, weakly , 13 6 of valued fields , 2 7 open conve x set , 13 9 of valued modules , 3 order compatible , see also compatibl e wit h increasing automorphism , 7 8 the orde r induced b y hf, 2 6 order complete , 3 0 infinitely smaller , 9 order completion , 3 0 infinitesimals, 1 7 order dense , 3 0 isomorphic a s ordere d skeletons , 1 0 order preserving , 8 isomorphic a s value d modules , 2 ordered module , 8 isomorphism homomorphism of , 8 of ordered systems , 2 ordered skeleton , 9 of valued modules , 2 ordered syste m isomorphism o f ordered systems , 2 Kaplansky, I. , 2 8 of modules, 2 left piecewise monotone , 13 8 •o-left exponential , 2 4 polynomial -u-left logarithm , 2 4 generalized x-polynomial , 13 9 to-left prelogarithm , 6 6 reduced, 2 8 (GA)-i>-left exponential , 3 4 poly normally bounded , 9 5 level, 9 7 positive infinit e elements , 1 7 levelled, 9 7 positive units , 5 0 lexicographic decompositio n power serie s field, 2 7 additive, 1 8 exponential-logarithmic multiplicative, 1 9 over To , 7 5 lexicographic product , 1 3 over (ro,cr) , 8 1 of exponentials, 2 5 precontraction, 56 , 12 1 lexicographic sum , 1 3 centrifugal, 12 1 lifting, 26 , 6 6 centripetal, 12 1 lifting property , 3 5 precontraction group , 12 0 lifts, 2 5 prelogarithm, 5 3 logarithm, 2 2 w-compatible, 5 3 ^-compatible, 2 3 w-left, 6 6 •u-left logarithm , 2 4 UHright, 6 6 logarithmic cross-section , 6 6 (GA)-prelogarithm, 5 4 surjective, 6 6 (Ti)-prelogarithm, 5 5 Logarithmico-Exponential functions , 9 4 exponential rank , 5 3 maximal exponential , 2 9 principal maximally value d X^-principal, 6 1 field, 2 7 X^-principal generate d b y g, 6 1 module, 4 ^-principal, 6 1 middle ^-principal generate d b y a , 6 1 v-middle exponential , 2 5 (^-principal, 6 1 (GAT)-'u-middle exponential , 3 4 Ge -principal generate d b y 7 , 6 1 minimum suppor t valuation , 3 , 2 7 principal exponential rank , 6 1 monomial, 11 4 principal conve x subgrou p generalized x-monomial , 13 9 generated b y g, 5 1 monic generalize d x _monomial, 13 8 principal conve x submodule, 8 multiplicative class , 5 2 principal conve x subring generate d b y a , 5 1 Multiplicative Lexicographi c Decompositio n principal exponentia l rank , 6 1 Theorem, 1 9 characterization, 6 3 principal final segment , 5 1 natural contraction , 4 5 principal ran k induced b y h, 4 5 of a field, 5 1 induced b y /, 4 5 of a group, 5 1 162 Index pseudo Cauch y sequence , 5 ultimately, 9 2 pseudo complete , 5 ultrametric inequality , 1 pseudo limit , 5 units group o f 1-units , 16 , 1 8 quasicut, 11 8 group o f positive units , 1 8 group o f units, 1 6 rank positive, 5 0 cr-rank, 8 3 exponential, 5 2 valuation of a value d group , 5 0 ^min, 3 , 2 7 of an ordere d field, 5 0 canonical, 2 7 principal coarser, 5 0 of a field, 5 1 convex, 16 , 4 9 of a group, 5 1 finer, 5 0 principal exponentia l rank , 6 1 on a field, 1 5 realize a (quasi)cut , 11 8 on a module, 1 reduced polynomial , 2 8 valuation basis , 4 representable valuation ideal , 1 6 P-representable, 14 2 valuation independent , 4 residue field, 1 6 over, 4 restricted analyti c function , 10 3 valuation preserving , 2 Riemann ("-function , 11 3 valuation ring , 1 6 right value group , 1 5 v-right exponential , 2 5 value set , 1 lu-right prelogarithm , 6 6 value se t preservin g extension , 12 4 (Tn)-i>-right exponential , 4 2 valued field, 1 5 (T)-f-right exponential , 3 4 henselian, 2 8 root close d fo r positiv e elements , 1 8 immediate extension , 2 7 maximally valued , 2 7 shift valued module , 1 induced b y a prelogarithm, 5 7 embedding o f valued modules , 2 to the right , 5 7 extension o f valued modules , 3 shift elements , 13 1 immediate extensio n of , 3 shift o f a quasicut , 11 8 isomorphism o f valued modules , 2 skeleton, 2 maximally valued , 4 isomorphic a s ordered skeletons , 1 0 ordered, 9 weakly o-minimal , 13 6 strong ^-logarithmic cross-section , 7 7 exponential group , 3 5 group exponential , 3 5 sum o f chains , 1 0 sum o f ordered system s o f modules, 1 0 supersequence, 13 1 support, 3 , 27, 6 7 surjective w;-logarithmi c cross-section , 6 6
Taylor axiom , 3 3 theorem Additive Lexicographic Decomposition, 1 8 Countable Cas e Characterization , 3 1 Holder's Theorem , 1 5 Multiplicative Lexicographi c Decomposi - tion Theorem , 1 9 topology i^-topology, 2 9 transcendental X-transcendental, 13 1 bounded x-transcendental , 13 3 transexponential, 9 5 truncation close d embedding , 8 7 List o f Notatio n
ei6/M, i v 1 (M,v) 1 v(M) 1
[r,{B(7);7er}] 2
[^7 | 7 el^}] 2 M7 2
MT 2 £(M, 7) 2 S(M) 2 5(7) 2 M TT (7, a;) 2
2 TT(7 . Z ) n76rR(7) 3 support(s) 3
076r£(7) 3
^min < J
U7er5(7) 3
H76r S(7 ) 3 #<{*i|ieJ}> 4 ^(M) 8
DX{M) 8
BX{M) 8 |x| 9 Z 9 R « 9 MR 9 r 9 vR 9
163 164 List o f Notatio n
MIIN 1 0
Aj + A 2 1 0
Sj_ IIS2 1 0
Uw 1 6
1 + I w 1 6 K>0 1 6 [a] 1 6 K 1 7 PK 1 7 v, VG 1 7 G<0, G >0, G^° 1 7 U>° 1 8
1 + IV 1 8 -v 1 9 v 1 9
^X ? -L^X 1 *^X ) *-*X 1 ^X 5 L^X £*J ~, >, < 2 0 (K,f) 2 2 hf 2 5 /LU/MU/R 2 5 hi 2 6 Rr 2 7
17 2 7 H(G)) 2 7 »min 2 7 k[[G}} 2 7 Neg(K) 2 7 k({G<0)) 2 7 Mon(K) 2 7 p(x)w 2 8
TK 3 0 List o f Notation 165
Kc 3 0
En(x) 3 3 (GA) 3 3 (T) 3 3 (T„) 3 3
Pn(x,y),T^(x,y),Qn(x,y) 3 7 X 44,12 0 Xf 4 5 49 (CO) 4 9 U>° 4 9 n 5 0
Gw 5 0 r^, 5 1 rfs 5 1 r* 5 1 WT 5 1 [a\ 5 2 (CE) 5 2 fw 5 2
Kf 5 2 (Ti) 5 3
Kt 5 3 Xt 5 6 Ce 5 6 ~V 5 7 [a]
~x< > ~C/ 58
[a]t 5 8
A.w , B„, 6 5 t%,tw,e% 6 6 o-Emb((1 + IW,-),(IW,+)) 6 6 hf 6 6 o-Emb (w(K), (K,+)\R W) 6 6 LK,L£ 6 6 X*,X£ 6 6 166 List o f Notatio n support(a) 6 7 £L M((r0)) 7 5 EL( LEjr{x) 9 8 Kan 10 3 Tan 10 3 TLE 10 4 •Fan 10 4 Tan(exp) 10 7 u 10 7 KV,K£ 10 9 LE?{logmo(x)) 11 0 R((t))LE 11 2 £cg 12 0 PCp, Vet 12 3 Px 12 4 Gb 12 9 Cut(6) 13 2 Ccg{G, x) 13 7 Oa, M a 13 8 X[a], xWi,--- ,a n] 13 9 G[a], G[ai,... , an] 13 9 Sf(b) 15 1