Logarithmic limit sets of real semi-algebraic sets

Transcription

1 Ahead of Prin DOI / advgeom Advances in Geomery c de Gruyer 20xx Logarihmic limi ses of real semi-algebraic ses Daniele Alessandrini (Communicaed by C. Scheiderer) Absrac. This paper is abou he logarihmic limi ses of real semi-algebraic ses, and, more generally, abou he logarihmic limi ses of ses definable in an o-minimal, polynomially bounded srucure. We prove ha mos of he properies of he logarihmic limi ses of complex algebraic ses hold in he real case. This includes he polyhedral srucure and he relaion wih he heory of non-archimedean fields, ropical geomery and Maslov dequanizaion. 1 Inroducion Logarihmic limi ses of complex algebraic ses have been sudied exensively. They firs appeared in Bergman s paper [3], and hen hey were furher sudied by Bieri and Groves in [4]. Recenly heir relaions wih he heory of non-archimedean fields and ropical geomery were discovered (see for example [17], [10] and [6]). They are now usually called ropical varieies, bu hey appeared also under he names of Bergman fans, Bergman ses, Bieri-Groves ses or non-archimedean amoebas. The logarihmic limi se of a complex algebraic se is a polyhedral complex of he same dimension as he algebraic se, i is described by ropical equaions and i is he image of an algebraic se over an algebraically closed non-archimedean field under he componen-wise valuaion map. The ools used o prove hese facs are mainly algebraic and combinaorial. In his paper we exend hese resuls o he logarihmic limi ses of real algebraic and semi-algebraic ses. The echniques we use o prove hese resuls in he real case are very differen from he ones used in he complex case. Our main ool is he cell decomposiion heorem, as we prefer o look direcly a he geomeric se, insead of using is equaions. In he real case, even if we resric our aenion o an algebraic se, i seems ha he algebraic and combinaorial properies of he defining equaions do no give enough informaion o sudy he logarihmic limi se. In he following we ofen need o ac on (R >0 ) n wih maps of he form ϕ A (x 1,..., x n ) = ( ) x a11 1 x a1n n, x a21 1 x a2n n,..., x an1 1 x ann n,

2 2 Daniele Alessandrini where A = (a ij ) is an n n marix. When he enries of A are no raional, he image of a semi-algebraic se is, in general no semi-algebraic. Acually, he only hing we can say abou images of semi-algebraic ses via hese maps is ha hey are definable in he srucure of he real field expanded wih arbirary power funcions. This srucure, usually denoed by R R, is o-minimal and polynomially bounded, and hese are he main properies we need in he proofs. Moreover, if S is a se definable in R R, hen he image ϕ A (S) is again definable, as he funcions x x α are definable for every α. If a srucure has he propery ha all power funcions wih every real exponen are definable, ha srucure is said o have field of exponens R. In his sense he caegory of semi-algebraic ses is oo small for our mehods. I seems ha he naural conex for he sudy of logarihmic limi ses is o fix a general expansion of he srucure of he real field ha is o-minimal and polynomially bounded, wih field of exponens R. For ses definable in such a srucure, he properies ha were known for he complex algebraic ses also hold. We can prove ha hese logarihmic limi ses are polyhedral complexes wih dimension less han or equal o he dimension of he definable se. Then we show how he relaion beween ropical varieies and images of varieies defined over non-archimedean fields, well known for algebraically closed fields, can be exended o he case of real closed fields. We give he noion of non-archimedean amoebas of semi-algebraic ses over non-archimedean fields, and also for ses definable in oher o-minimal srucures on non-achimedean fields. We show ha he logarihmic limi se of a definable subse of R n is he non-archimedean amoeba of an exension of he definable se o a real closed non-archimedean field. We also sudy he relaions beween non-archimedean amoebas and pachworking families of definable ses. Noe ha his noion of non-archimedean amoebas of definable ses generalizes he noion of non-archimedean amoebas of semi-linear ses ha has been used in [8] o sudy ropical polyopes. An analysis of he defining equaions and inequaliies is carried ou, showing ha he logarihmic limi se of a closed semi-algebraic se can be described applying he Maslov dequanizaion o a suiable formula defining he semi-algebraic se. Our moivaion for his work comes from he sudy of Teichmüller spaces and, more generally, of spaces of geomeric srucures on manifolds. In he papers [1] and [2] we presen a consrucion of compacificaions using he logarihmic limi ses. The properies of logarihmic limi ses we prove here will be used in [1] o describe he compacificaion. For example, he fac ha logarihmic limi ses of real semi-algebraic ses are polyhedral complexes will provide an independen consrucion of he piecewise linear srucure on he Thurson boundary of Teichmüller spaces. Moreover he relaions wih ropical geomery and he heory of non-archimedean fields will be used in [2] for consrucing a geomeric inerpreaion of he boundary poins. In Secion 2 we define logarihmic limi ses for general subses of (R >0 ) n, and we recall some preliminary noions of model heory and o-minimal geomery ha we will use in he following, mos noably he noion of regular polynomially bounded srucures. In Secion 3 we prove ha logarihmic limi ses of definable ses in a regular polynomially bounded srucure are polyhedral complexes wih dimension less han or equal o he

3 Logarihmic limi ses of real semi-algebraic ses 3 dimension of he definable se, and we provide a local descripion of hese ses. The main ool we use in his secion is he cell decomposiion heorem. In Secion 4 we consider a special class of non-archimedean fields: he Hardy fields of regular polynomially bounded srucures. These are non-archimedean real closed fields of rank one exending R, wih a canonical real valued valuaion and residue field R. The elemens of hese fields are germs of definable funcions, hence hey have beer geomeric properies han he fields of formal series usually employed in ropical geomery. The images, under he componen-wise valuaion map, of definable ses in he Hardy fields are relaed wih he logarihmic limi ses of real definable ses, and wih he limi of real pachworking families. In Secion 5 we compare he consrucion of his paper wih oher known consrucions. We show ha he logarihmic limi ses of complex algebraic ses are only a paricular case of he logarihmic limi ses of real semi-algebraic ses, and he same happens for he limi of complex pachworking families. Hence our mehods provide an alernaive proof (wih a opological flavor) for some known resuls abou complex ses. We also compare he logarihmic limi ses of real algebraic ses wih he consrucion of Posiive Tropical Varieies (see [18]). Even if in many examples hese wo noions coincide, we show some examples where hey differ. In Secion 6 we show how he consrucion of Maslov dequanizaion provides a relaion beween logarihmic limi ses of semi-algebraic ses and he ropical semifield. We show ha for every closed semi-algebraic se here exiss a defining formula such ha he ropicalizaion of ha formula defines he logarihmic limi se. This resul is he analog, for real semi-algebraic ses, of he exisence of a ropical basis for complex algebraic ses. 2 Preliminaries 2.1 Noaion. If x R n we will denoe is coordinaes by x 1,..., x n. If ω N n we will use he muli-index noaion for powers: x ω = x ω xωn n. We will consider also powers wih real exponens, if he base is posiive, hence if x (R >0 ) n and ω R n we will wrie x ω = x ω xωn n. If (s(k)) is a sequence in R n, we will denoe is k-h elemen by s(k) R n. The subscrip noaion will be reserved for he coordinaes s i (k) R. Given a real number α > 1, we will denoe by Log α he componen-wise logarihm map, and by Exp α is inverse: Log α : (R >0 ) n (x 1,..., x n ) (log α (x 1 ),..., log α (x n )) R n Exp α : R n (x 1,..., x n ) (α x1,..., α xn ) (R >0 ) n We define a noion of limi for every one-parameer family of subses of R n. Suppose ha for all (0, ε) we have a se S R n. We can consruc he deformaion D(S ) = {(x, ) R n (0, ε) x S }. We denoe by D(S ) he closure of D(S ) in R n [0, ε), hen we define lim 0 S = π(d(s ) R n {0}) R n,

4 4 Daniele Alessandrini Figure 1. If V = {(x, y) (R >0 ) 2 y = sin x + 2, x 5} (lef picure), hen A 0 (V ) = {(x, y) R 2 y = 0, x 0} (righ picure). where π : R n [0, ε) R n is he projecion on he firs facor. This limi is well defined for every family of subses of R n. Proposiion 2.1. The se S = lim 0 S is a closed subse of R n. A poin y is in S if and only if here exis a sequence (y(k)) in R n and a sequence ((k)) in (0, ε) such ha ((k)) 0, (y(k)) y and k N : y(k) S (k). 2.2 Logarihmic limi ses of general ses. Given a se V (R >0 ) n and a number (0, 1), he amoeba of V is A (V ) = Log 1 (V ) = 1 log e () Log e(v ) R n. The limi of he amoebas (in he sense of subsecion 2.1) is he logarihmic limi se of V : A 0 (V ) = lim 0 A (V ). Some examples of logarihmic limi ses are in Figures 1 and 2. Proposiion 2.2. Given a se V (R >0 ) n he following properies hold: (1) The logarihmic limi se A 0 (V ) is closed and y A 0 (V ) if and only if here exis a sequence (x(k)) in V and a sequence ((k)) in (0, 1) such ha ((k)) 0 and ( ) Log 1 (k) (x(k)) y. (2) The logarihmic limi se A 0 (V ) is a cone in R n. (3) We have 0 A 0 (V ) if and only if V. Moreover, A 0 (V ) = {0} if and only if V is relaively compac in (R >0 ) n and non-empy. (4) If W R n we have A 0 (V W ) = A 0 (V ) A 0 (W ) and A 0 (V W ) A 0 (V ) A 0 (W ).

5 Logarihmic limi ses of real semi-algebraic ses 5 Figure 2. If V = {(x, y) (R >0 ) 2 x 2 y x} (lef picure), hen A 0 (V ) = {(x, y) R 2 2x y 1 2x} (righ picure). Proof. The firs asserion is simply a resaemen of Proposiion 2.1. For he second one, we wan o prove ha if λ > 0 and y A 0 (V ), hen λ 1 y A 0 (V ). There exis a sequence (x(k)) in V and a sequence ((k)) in (0, 1) such ha ((k)) 0 and ( Log 1 (x(k)) ) y. Consider he sequences (x(k)) and ( (k) λ). Now (k) Log 1 (k) λ (x(k)) = 1 ( λ log ) Log e(x(k)) = λ 1 1 e (k) log e ((k)) Log e(x(k)) and his sequence converges o λ 1 y. The hird and fourh asserions are rivial. Given a closed cone C R n, here always exiss a se V (R >0 ) n such ha C = A 0 (V ), simply ake V = Log 1 e (C). Then A (V ) = C for all. If we wan o find more properies of logarihmic limi ses, we need some assumpions on he se V, for example ha V is semi-algebraic or, more generally, definable in an o-minimal polynomially bounded srucure, as in Secion 3. Le A = (a ij ) GL n (R). The marix A acs on R n in he naural way and, via conjugaion wih he map Log e, i acs on (R >0 ) n. Explicily, i induces he maps A : R n R n and A : (R >0 ) n (R >0 ) n : A(x) = A(x 1,..., x n ) = (a 11 x a 1n x n,..., a n1 x a nn x n ) A(x) = Exp e A Log e (x) = (x a11 1 xa12 2 x a1n n If V (R >0 ) n and B GL n (R), hen B(A 0 (V )) = A 0 (B(V )).,..., x an1 1 x an2 2 x ann n ) Lemma 2.3. (0,..., 0, 1) A 0 (V ) if and only if here exiss a sequence (y(k)) in V such ha (y n (k)) 0 and N N : k 0 N : k > k 0 : i {1,..., n 1} : y n (k) < (y i (k)) N and y n (k) < (y i (k)) N.

6 6 Daniele Alessandrini Proof. Suppose (0,..., 0, 1) A 0 (V ), hen by Proposiion 2.2 here exiss a sequence (y(k)) in V and a sequence ((k)) in (0, 1) such ha ((k)) 0 and ( Log 1 (y(k)) ) (k) (0,..., 0, 1). This means ha ( ) { 1 log e ((k)) log 1 if i = n e(y i (k)) 0 if i {1,..., n 1} Now ((k)) 0 hence ( ( 1 log e ((k))) 0, loge (y n (k)) ), (y n (k)) 0 and ( ) loge (y i (k)) i {1,..., n 1} : 0. log e (y n (k)) Hence ε > 0 : k 0 : k > k 0 : i < n : (y n (k)) ε < y i (k) < (y n (k)) ε. We conclude by reversing he inequaliies and choosing ε = 1 N. Conversely, if (y(k)) has he saed propery, hen ( Log e (y(k)) ) (where is he sandard Euclidean norm in R n ). I is possible o choose ((k)) such ha ((k)) 0 and Log 1 (k) (y(k)) = 1. Up o subsequences, he sequence ( Log 1 (k) (y(k)) ) converges o a poin ha, by reversing he calculaions in firs par of he proof, is (0,..., 0, 1). Hence (0,..., 0, 1) A 0 (V ). Corollary 2.4. If here exiss a sequence (x(k)) in V such ha (x(k)) (a 1,..., a n 1, 0), where a 1,..., a n 1 > 0, hen (0,..., 0, 1) A 0 (V ). The converse is no rue in general (see Figure 3). We will see in Theorem 3.6 ha if V is definable in an o-minimal and polynomially bounded srucure, he converse of he corollary becomes rue. A sequence (b(k)) in (R >0 ) n is in sandard posiion in dimension m if, for g = n m, we have (b(k)) b = (b 1,..., b g, 0,..., 0), wih b 1,..., b g > 0 and N N : k 0 : k > k 0 : i {g + 1,..., n 1} : b i+1 (k) < (b i (k)) N. Figure 3. If V = {(x, y) (R >0 ) 2 y = e 1 x 2 } (lef picure), hen A 0 (V ) = {(x, y) R 2 y = 0, x 0 or x = 0, y 0} (righ picure).

7 Logarihmic limi ses of real semi-algebraic ses 7 Lemma 2.5. Le (a(k)) be a sequence in (R >0 ) n such ha (a(k)) a = (a 1,..., a h, 0,..., 0), wih h < n and a 1,..., a h > 0. There exis a number m n h, a subsequence (again denoed by (a(k))) and a linear map A : R n R n such ha he sequence (b(k)) = ( A(a(k)) ) (R >0 ) n is in sandard posiion in dimension m. Proof. By inducion on n. For n = 1 he saemen is rivial. Suppose ha he saemen holds for n 1. Consider he logarihmic image of he sequence: ( Log e (a(k)) ). Up o exracing a subsequence, he sequence ( Log e (a(k)) Log e (a(k)) ) converges o a uni vecor v = (0,..., 0, v h+1,..., v n ). There exiss a linear map B, acing only on he las n h coordinaes, sending v o (0,..., 0, 1). By Lemma 2.3, he map B sends (a(k)) o a sequence (b(k)) such ha (b n (k)) 0 and N N : k 0 : k > k 0 : i {1,..., n 1}: b n (k) < (b i (k)) N and b n (k) < (b i (k)) N (1) As B acs only on he las n h coordinaes, (b i (k)) a i 0 for i {1,..., h}. Up o subsequences we can suppose ha for every i {h + 1,..., n 1} one of he following hree possibiliies occurs: (b i (k)) 0, (b i (k)) b i 0, (b i (k)) +. Up o a change of coordinaes wih maps of he form B i (x 1,..., x i,..., x n ) = (x 1,..., x i,..., x n ) B i (x 1,..., x i,..., x n ) = (x 1,..., x 1 i,..., x n ), we can suppose ha for every i {h + 1,..., n 1} eiher (b i (k)) 0 or (b i (k)) b i 0. Up o reordering he coordinaes, we can suppose ha here exiss g h such ha for i {1,..., g}: (b i (k)) b i 0 and for i > g, (b i (k)) 0. Now consider he projecion on he firs n 1 coordinaes: π : R n R n 1. By he inducive hypohesis here exiss a linear map C : R n 1 R n 1 sending he sequence (π(b(k))) o a sequence (c(k)) saisfying N N : k 0 : k > k 0 : i {g + 1,..., n 2} : c i+1 (k) < (c i (k)) N (2) The composiion of B and a map ha preserves he las coordinae and acs as C on he firs ones is he linear map we are searching for. For i {g + 1,..., n 2}, he inequaliy follows from (2), for i = n 1, i follows from (1). The basic cone defined by he vecor N = (N 1,..., N n 1 ) N n 1 is B N = {x R n i : x i 0 and i < n : x i+1 N i x i }. Noe ha if N = (N 1,..., N n 1 ), wih i : N i N i, hen B N B N. The exponenial basic cone in (R >0 ) n defined by he vecor N = (N 1,..., N n 1 ) N n 1 and he scalar h > 0 is he se E N,h = { x R n i : 0 < x i h and i < n : x i+1 x i N i }. Lemma 2.6. The following easy facs abou basic cones hold:

8 8 Daniele Alessandrini (1) The logarihmic limi se of an exponenial basic cone is a basic cone: A 0 (E N,h ) = B N. (2) If (b(k)) is a sequence in (R >0 ) n in sandard posiion in dimension n, and E N,h is an exponenial basic cone, hen for large enough k, b(k) E N,h. 2.3 Definable ses in o-minimal srucures. In his subsecion we recall noaion and some definiions of model heory and o-minimal geomery we will use laer, see [9] and [19] for deails. Given a se of symbols S (see [9, Chaper II, Definiion 2.1]), we denoe by L S he corresponding firs order language (see [9, Chaper II, Definiion 3.2]). If S is an expansion of a se of symbols S we will wrie S S. For example, for he heory of real closed fields one can use he se of symbols of ordered semirings: OS = ({ }, {+, }, ) or he se of symbols of ordered rings OR = ({ }, {+,, }, {0, 1}), which is an expansion of OS. In he following we will use hese ses of symbols and some of heir expansions. We usually will denoe an S-srucure by M = (M, a), where M is a se, and a is he inerpreaion (see [9, Chaper III, Definiion 1.1]). Given an S-srucure M = (M, a), and an L S -formula ϕ wihou free variables, we will wrie M ϕ if M saisfies ϕ (see [9, Chaper III, Definiion 3.1]). A real closed field can be defined as an OS- or an OR-srucure saisfying a suiable infinie se of firs order axioms. The naural OS-srucure on R will be denoed by R. If M = (M, a) is an S-srucure, and S is an expansion of S, an S -srucure (M, a ) is an expansion of he S-srucure (M, a) if a resriced o he symbols of S is equal o a. If M N, an S-srucure N = (N, b) is an exension of an S-srucure M = (M, a) if for all s S, b(s) M = a(s). A definable subse of M n is a se ha is defined by an L S -formula ϕ(x 1,..., x n, y 1,..., y m ) and by parameers a 1,... a m M, and a definable map is a map whose graph is definable. For example if M is an OS- or an OR-srucure saisfying he axioms of real closed fields, he definable ses are he semi-algebraic ses, and he definable maps are he semi-algebraic maps. Le S be an expansion of OS, and le M be an S-srucure saisfying he axioms of he real closed fields. The S-srucure M is said o be o-minimal if he definable subses of M are he finie unions of poins and inervals (bounded and unbounded). See also [7] and [19]. Le S be an expansion of OS, and le R = (R, a) be an o-minimal S-srucure. The S-srucure R is called polynomially bounded if for every definable funcion f : R R here exiss a naural number N such ha for every sufficienly large x, f(x) x N. See also [12]. In [13] i is shown ha, in his case, if f : R R is definable and no ulimaely 0, here exis r, c R, c 0, such ha f(x) lim x + x r = c. The se of all such r is a subfield of R, called he field of exponens of R. For example he OR-srucure R is polynomially bounded wih field of exponens Q.

9 Logarihmic limi ses of real semi-algebraic ses 9 If Λ R is a subfield, we can consruc an expansion of S and R by adding he power funcions wih exponens in Λ. We expand S o S Λ by adding a funcion symbol f λ for every λ Λ, and we expand R o an S Λ -srucure R Λ inerpreing he funcion symbol f λ by he funcion ha is x x λ for posiive numbers and x 0 on negaive ones. The srucure R Λ is again o-minimal, as is definable ses are definable in he srucure R expanded by adding he exponenial funcion e x, which is o-minimal by [16]. Suppose ha he expansion of R consruced by adding he family of funcions {x r [1,2] } is polynomially bounded, hen r Λ RΛ is oo (see [14]). For example if he srucure R expanded by adding he resriced exponenial funcion e x [0,1] is polynomially bounded, hen R Λ is oo. In he following we will work wih o-minimal, polynomially bounded srucures R expanding R, wih he propery ha R R is polynomially bounded. We will call such srucures regular polynomially bounded srucures. One example of regular polynomially bounded srucure is R an, he real numbers wih resriced analyic funcions, see [20] for deails. This srucure has field of exponens Q, while R R an has field of exponens R. As R an is an expansion of R, also R R is polynomially bounded, hence R is a regular polynomially bounded srucure. Oher examples of regular polynomially bounded srucures are he srucure R an of he real field wih convergen generalized power series (see [22]), he field of real numbers wih mulisummable series (see [23]) and he srucures defined by a quasianalyic Denjoy Carleman class (see [15]). 3 Logarihmic limi ses of definable ses 3.1 Some properies of definable ses. Le R be an o-minimal and polynomially bounded expansion of R. Lemma 3.1. For every definable funcion f : (R >0 ) n R >0, here is an exponenial basic cone C and N N such ha f C (x 1,..., x n ) (x n ) N. Proof. Fix an exponenial basic cone C (R >0 ) n. By he Łojasiewicz inequaliy (see [21, 4.14]) here exis N N and Q > 0 such ha Qf C (x 1,..., x n ) (x n ) N. The asserion follows by choosing an exponen bigger han N and a suiable exponenial basic cone smaller han C. Lemma 3.2. Every cell decomposiion of (R >0 ) n has a cell conaining an exponenial basic cone. Proof. This proof is based on he cell decomposiion heorem, see [19, Chaper 3] for deails. We proceed by inducion on n. For n = 1, he saemen is rivial. Suppose he lemma rue for n. If {C i } is a cell decomposiion of (R >0 ) n+1, and if π : (R >0 ) n+1 (R >0 ) n is he projecion on he firs n coordinaes, hen {π(c i )} is a

10 10 Daniele Alessandrini cell decomposiion of (R >0 ) n, hence, by inducion, i conains an exponenial basic cone D of (R >0 ) n. Then π 1 (D) (0, 1] conains a cell of he form E = {( x, x n+1 ) x D, 0 < x n+1 < f( x)}, where x = (x 1,..., x n ) and f : D (0, 1] is definable. By he previous lemma, here is an exponenial basic cone D D and N N such ha f D ( x) (x n ) N. Hence E conains he exponenial basic cone { ( x, xn+1 ) x D, 0 < x n+1 (x n ) N}. Corollary 3.3. Le V (R >0 ) n be definable in R, and suppose ha V conains a sequence (x(k)) in sandard posiion in dimension n. Then V conains an exponenial basic cone. Proof. Le {C i } be a cell decomposiion of (R >0 ) n adaped o V. By he previous lemma, one of he cells conains an exponenial basic cone D. By Lemma 2.6, if k is sufficienly large, (x(k)) D, hence D V. Corollary 3.4. Le V (R >0 ) 2 be definable in R, and suppose ha exiss a sequence (x(k)) in V such ha (x(k)) 0 and N N : k 0 : k > k 0 : x 2 (k) < (x 1 (k)) N. Then here exis h 0 > 0 and M N such ha { x R 2 0 < x 1 < h 0 and 0 < x 2 < (x 1 ) M } V. Proof. This is precisely he previous corollary wih n = 2. Lemma 3.5. Le V (R >0 ) n be definable in R, and suppose ha here exiss a sequence (x(k)) in V, an ineger m {1,..., n} and, for g = n m, posiive numbers a 1,..., a g > 0 such ha (x(k)) (a 1,..., a g, 0,..., 0), and such ha N N : k 0 : k > k 0 : i {g + 1,..., n 1} : x n (k) < (x i (k)) N. Then for every ε > 0 here exis a sequence (y(k)) in V and posiive real numbers b 1,... b n 1 > 0 such ha (y(k)) (b 1,... b n 1, 0) and for all i {1,... g} we have b i a i < ε. Proof. If n = 2 he saemen follows by Corollary 3.4. By inducion on n we suppose he saemen o be rue for definable ses in R n wih n < n. We spli he proof in wo cases, when m < n and when m = n. If m < n, fix an ε > 0, smaller han every a i, and consider he parallelepiped c ε = {(z 1,..., z n ) R n z1 a 1 < 1 2 ε,..., z g a g < 1 2 ε }.

11 Logarihmic limi ses of real semi-algebraic ses 11 Le π : R n R m be he projecion on he las m coordinaes. The se π(v c ε ) is definable in R m, he sequence (π(x(k))) saisfies he hypoheses of he lemma, hence, by inducion, here exiss a sequence (z(k)) π(v c ε ) converging o he poin (b g+1,..., b n 1, 0). Le (y(k)) be a sequence such ha y(k) π 1 (z(k)). We can exrac a subsequence (called again (y(k))) such ha (y(k)) (b 1,..., b n 1, 0) where for all i {1,... g} we have b i a i 1 2 ε. If m = n > 2, hen (x(k)) 0. The sequence ( (x 1(k),...,x (x 1(k),...,x n 1(k)) ) is conained in he uni sphere S n 2, and, up o subsequences, we can suppose ha i converges o a uni vecor v = (v 1,..., v n 1 ) (R 0 ) n 1. Up o reordering, v = (v 1,..., v h, 0,..., 0), wih v 1,..., v h > 0. We can choose an α > 0 small enough such ha he closure of he open cone { C v (α) = y R n (y1,..., y n 1 ), v } (y 1,..., y n 1 ) v > cos α inersecs he coordinae hyperplane {y y 1 = 0} only in 0. Le π : R n R n 1 he projecion on he las n 1 coordinaes. The se π(v C v (α)) is definable in R n 1, he sequence (π(x(k))) saisfies he hypoheses of he lemma, hence, by inducion, here exiss a sequence (z(k)) in π(v C v (α)) converging o he poin (b 2,..., b n 1, 0), wih b 2,..., b n 1 > 0. Le (y(k)) be a sequence such ha y(k) π 1 (z(k)). Up o subsequences, (y(k)) (b 1,..., b n 1, 0). To see ha b 1 > 0, noe ha y(k) C v (α), whose closure inersecs {y y 1 = 0} only in 0. Hence if, by conradicion, we have ha (y 1 (k)) 0, hen (y(k)) 0. Bu b 2,..., b n 1 > 0, a conradicion. 3.2 Polyhedral srucure. Le V (R >0 ) n be a definable se in an o-minimal and polynomially bounded srucure. Our main objec of sudy is A 0 (V ), he logarihmic limi se of V. Suppose ha R has field of exponens Λ R. Given a marix B GL n (Λ), he se B(A 0 (V )) is he logarihmic limi se of B(V ). The componens of B 1 are all definable in R because heir exponens are in Λ, hence he se B(V ) is again definable. Theorem 3.6. Le V (R >0 ) n be a se definable in an o-minimal and polynomially bounded srucure. The poin (0,..., 0, 1) is in A 0 (V ) if and only if here exiss a sequence (x(k)) in V such ha (x(k)) (a 1,..., a n 1, 0), where a 1,..., a n 1 > 0. Proof. If here exiss such an (x(k)), hen i is obvious ha (0,..., 0, 1) A 0 (V ). Vice versa, if (0,..., 0, 1) A 0 (V ), hen by Lemma 2.3 here exiss a sequence (y(k)) in V such ha (y n (k)) 0 and N N : k 0 N : k > k 0 : i {1,..., n 1} : y n (k) < (y i (k)) N and y n (k) < (y i (k)) N. Up o subsequences we can suppose ha for all i {1,..., n 1} one of he following hree possibiliies occur: (y i (k)) 0, (y i (k)) a i 0, (y i (k)) +. Up o a change of coordinaes wih maps of he form B i (x 1,..., x i,..., x n ) = (x 1,..., x i,..., x n ), we can suppose ha for all i {1,..., n 1} eiher (y i (k)) 0 or (y i (k)) a i 0. Then we can apply Lemma 3.5, and we are done.

12 12 Daniele Alessandrini Now we suppose ha R is a regular polynomially bounded srucure, or, equivalenly, ha R has field of exponens R. Le x A 0 (V ). We wan o describe a neighborhood of x in A 0 (V ). To do his, we choose a map B GL n (R) such ha B(x) = (0,..., 0, 1). Now we only need o describe a neighborhood of (0,..., 0, 1) in A 0 (B(V )). As logarihmic limi ses are cones, we only need o describe a neighborhood of 0 in H = {(x 1,..., x n 1 ) R n 1 (x 1,..., x n 1, 1) A 0 (B(V ))}. We define a one-parameer family and is limi (in he sense of subsecion 2.1): W = {(x 1,..., x n 1 ) (R >0 ) n 1 (x 1,..., x n 1, ) B(V )}, W = ( lim 0 W ) (R>0 ) n 1. The se W is a definable subse of (R >0 ) n 1. Is logarihmic limi se is denoed, as usual, by A 0 (W ) R n 1. By he previous heorem, as (0,..., 0, 1) A 0 (B(V )), W is no empy, hence 0 A 0 (W ). We wan o prove ha here exiss a neighborhood U of 0 in R n 1 such ha A 0 (W ) U = H U or, in oher words, ha A 0 (W ) H is a neighborhood of 0 boh in A 0 (W ) and H. This will be achieved in Theorem A flag in R n is a sequence (V 0, V 1,..., V h ), h n, of subspaces of R n such ha V 0 V 1 V h R n and dim V i = i. We use flags o encode he direcion in which a sequence converges o a poin, in he following sense: we say ha a sequence (x(k)) in R n converges o he poin y along he flag (V 1, V 2,..., V h ) if (x(k)) y, for k : x(k) y V h \V h 1 and i {0,..., h 2}, he sequence (π i (x(k) y)) converges o he poin π i (V i+1 ), where π i : V h \ V i P(V h /V i ) is he canonical projecion on he projecive space P(V h /V i ). Lemma 3.7. For all sequences (x(k)) in R n converging o a poin y, here exiss a flag (V 0,..., V h ) and a subsequence of (x(k)) converging o y along (V 0,..., V h ). Proof. I follows from he compacness of P(V h /V i ). Lemma 3.8. Le x(k) H be a sequence converging o 0. Then a leas one of is poins is in A 0 (W ) H. In paricular A 0 (W ) H conains a neighborhood of 0 in H. Proof. By Lemma 3.7, we can exrac a subsequence, again denoed by x(k), converging o zero along a flag (V 0, V 1,..., V h ) in R n 1. Up o a linear change of coordinaes, we can suppose ha his flag is given by ({0}, Span(e n 1 ), Span(e n 2, e n 1 ),..., Span(e n h,..., e n 1 )). Hence for i {1,..., n h 1} we have x i (k) = 0. Again by exracing a subsequence and by a change of coordinaes wih maps of he form B i (x 1,..., x i,..., x n ) = (x 1,..., x i,..., x n ), wih i {n h,..., n 1}, we can suppose ha for all such i, x i (k) < 0. By Proposiion 2.2, as H { 1} A 0 (B(V )), for every poin x(k) here exiss a sequence (y(k, l)) in B(V ) and a sequence ((k, l)) in (0, 1) such ha ((k, l)) 0

13 Logarihmic limi ses of real semi-algebraic ses 13 and ( Log 1 (y(k, l)) ) (x(k), 1). By Theorem 3.6 we can choose (y(k, l)) such (k,l) ha (y(k, l)) a(k), wih a i (k) > 0 for i {1,..., n h 1}, and a i (k) = 0 for i {n h,..., n}. Up o a change of coordinaes wih maps of he form B i (x 1,..., x i,..., x n ) = (x 1,..., x i,..., x n ), wih i {1,..., n h 1}, we can suppose ha he sequence a(k) is bounded and ha, up o subsequences, i converges o a poin a, wih a i = 0 for i {n h,..., n}. Le π : R n R n h 1 be he projecion on he firs n h 1 coordinaes. Then π(a(k)) (R >0 ) n h 1. By Lemma 2.5 we can suppose ha π(a(k)) is in sandard posiion, i.e. a 1,..., a g > 0, a g+1 = = a n = 0 and N N : k 0 : k > k 0 : i {g + 1,..., n h 2} : a i+1 (k) < (a i (k)) N. From he sequences y(k, l), we exrac a diagonal subsequence z(k) in he following way. For every k, he sequence y(k, l) converges o a(k) = (a 1 (k),..., a n h 1 (k), 0,..., 0). As Log 1 (k,l) (y(k, l)) (x(k), 1) = (0,..., 0, x n h (k),..., x n 1 (k), 1), for all i {n h,..., n 1} we have log e (y n (k, l)) log e (y i (k, l)) 1 x i (k). For every k, we can choose an l 0 (k) such ha log e (y 1 i(k,l 0(k))) x i(k) (1) i {n h,..., n 1} : log e (yn(k,l0(k))) (2) y(k, l 0 (k)) a(k) < 1 k. < 1 k, We define z(k) = y(k, l 0 (k)). Now (z(k)) a = (a 1,..., a g, 0,..., 0) and, as x(k) (0,..., 0) along he flag (V 0,..., V h ), we have N N : k 0 : k > k 0 : i {g + 1,..., n 1} : z i+1 (k) < (z i (k)) N. Le r be smaller han every a 1,..., a g. Consider he parallelepiped c r = {(z 1,..., z n ) R n z1 a 1 < 1 2 r,..., z g a g < 1 2 r }. Le π : R n R n g be he projecion on he las n g coordinaes. The se π(b(v ) c r ) is definable in R n g, and he sequence π(z(k)) saisfies he hypoheses of Corollary 3.3, hence π(b(v ) c r ) conains an exponenial basic cone, hence π(w { 1} c r ) also conains one. This means ha A 0 (π(w { 1}) c r ) conains a basic cone. Hence also A 0 (W { 1} c r ) conains his cone, and also A 0 (W ). By Lemma 2.6, a leas one of he poins z(k) is in his cone. Lemma 3.9. Le x A 0 (W ). Then he number r(x) = sup{r λx H, if 0 λ r} is sricly posiive.

14 14 Daniele Alessandrini Proof. Le x A 0 (W ). By a linear change of coordinaes, we can suppose ha x = (0,..., 0, 1) R n 1. By Theorem 3.6 here is a sequence (x(k)) in W converging o he poin (a 1,..., a n 2, 0), wih a 1,..., a n 2 > 0. As W is he limi of he family W, for every k here is a sequence (y(k, l)) in B(V ) converging o (x(k), 0). We can consruc a diagonal sequence (z(k)) in B(V ) in he following way: for every k we can choose an l 0 (k) such ha y(k, l 0 (k)) (x(k), 0) < (x n 1 (k)) k. Then we se z(k) = y(k, l 0 (k)). The sequence z(k) converges o he poin (a 1,..., a n 2, 0, 0). Le r be smaller han any of he a 1,..., a n 2. Consider he parallelepiped c r = {(z 1,..., z n ) R n z1 a 1 < 1 2 r,..., z n 2 a n 2 < 1 2 r }. Le π : R n R 2 be he projecion on he las 2 coordinaes. The se π(b(v ) c r ) is a definable subse of R 2, and he sequence π(z(k)) saisfies he hypoheses of Corollary 3.4, hence π(b(v ) c r ) conains an exponenial basic cone. This means ha here exiss a number r > 0 such ha {(0,..., 0, z) r z 0} H. Theorem Le V (R >0 ) n be a se definable in a regular polynomially bounded srucure. Le x A 0 (V ) and choose a map B GL n (R) such ha B(x) = (0,..., 0, 1). We recall ha H = {(x 1,..., x n 1 ) R n 1 (x 1,..., x n 1, 1) A 0 (B(V ))} W = {(x 1,..., x n 1 ) (R >0 ) n 1 (x 1,..., x n 1, ) B(V )} ( ) W = lim W (R >0 ) n 1. 0 Then here exiss a neighborhood U of 0 in R n 1 such ha A 0 (W ) U = H U. Proof. We will prove ha A 0 (W ) H is a neighborhood of 0 boh in A 0 (W ) and in H. The previous lemma implies ha if (x(k)) is a sequence in H converging o 0, hen a leas one of is poins is in A 0 (W ), hence A 0 (W ) H is a neighborhood of 0 in H. To prove ha A 0 (W ) H is also a neighborhood of 0 in A 0 (W ), we only need o prove ha if r is he funcion defined in Lemma 3.9, here exiss an ε > 0 such ha x A 0 (W ) S n 2 : r(x) > ε. Bu his is rue, because we already know ha A 0 (W ) H is a neighborhood of 0 in H. Theorem Le V (R >0 ) n be a se definable in a regular polynomially bounded srucure. The logarihmic limi se A 0 (V ) is a polyhedral complex. Moreover, if dim V = m, hen dim A 0 (V ) m.

15 Logarihmic limi ses of real semi-algebraic ses 15 Proof. By inducion on n. For n = 1 he saemen is rivial, as a cone in R is a polyhedral se, and every zero dimensional definable se is compac, hence is logarihmic limi se is a poin. Suppose he saemen is rue for n 1. For every x A 0 (V ) here is a linear map B sending x o (0,..., 0, 1). The saemen in [21, 4.7] implies ha he definable se W (R >0 ) n has dimension less han or equal o m 1, hence A 0 (W ) is a polyhedral se of dimension less han or equal o m 1 (by he inducive hypohesis). By he previous heorem a neighborhood of he ray {λx λ 0} in A 0 (V ) is he cone over a neighborhood of 0 in A 0 (W ), hence i is a polyhedral complex of dimension less han or equal o m. By compacness of he sphere S n 1, A 0 (V ) can be covered by a finie number of such neighborhoods, hence i is a polyhedral complex of dimension less han or equal o m. Noe ha he saemen abou he dimension can be false for a general se. See Figure 4 for an example. Moreover, i is no possible o give more han an inequaliy, as for every s m i is always possible o find a semi-algebraic se V (R >0 ) m such ha dim V = m and dim A 0 (V ) = s. For example ake he parallelepiped V = [1, 2] m s (R >0 ) s (R >0 ) m, wih A 0 (V ) = {0} m s (R >0 ) s. I is also possible o find counerexamples of his kind where V is he inersecion of (R >0 ) m+1 wih an algebraic hypersurface. For example le S m s (R >0 ) m s+1 be he sphere wih cener (2,..., 2) and radius 1, hen V = S m s (R >0 ) s (R >0 ) m+1 has dimension m, bu A 0 (V ) = {0} m s+1 (R >0 ) s has dimension s. I is also possible o find a semi-algebraic se V ha is he inersecion of (R >0 ) n wih an irreducible pure-dimensional smooh hypersurface, and such ha is logarihmic limi se A 0 (V ) is no pure-dimensional, see for example Figure 5. Noe ha he produc V S h, wih S h he sphere wih cener (2,..., 2) and radius 1 as above, is again he inersecion of (R >0 ) n+h+1 wih an irreducible pure-dimensional smooh variey, and is logarihmic limi se is lower dimensional and no pure-dimensional. Figure 4. If V = {(x, y) (R >0 ) 2 y = sin 1 x } (lef picure), hen A 0(V ) = {(x, y) R 2 y 0, x 0 or x 0, y = x} (righ picure).

16 16 Daniele Alessandrini Figure 5. If V = {(x, y, z) (R >0 ) 3 x = (y 1) 2 +(z 1) 2 } (lef picure), hen A 0 (V ) conains a ray along he direcion ( 1, 0, 0) ha is no conained in any wo-dimensional face, and a wo-dimensional par made up of hree infinie wo-dimensional faces in he half-space x 0 (righ picure). 4 Non-Archimedean descripion 4.1 The Hardy field. Le S be a se of symbols expanding OS, and le R = (R, a) be an o-minimal S-srucure expanding R (see subsecion 2.3 for definiions). The Hardy field of R can be defined in he following way. If f, g : R >0 R are wo definable funcions, we say ha hey have he same germ near zero, and we wrie f g, if here exiss an ε > 0 such ha f (0,ε) = g (0,ε). The Hardy field can be defined as he se of germs of definable funcions near zero: H(R) = {f : R >0 R f definable }/. We will denoe by [f] he germ of a funcion f. For every elemen a R, he consan funcion wih value a defines a germ ha is idenified wih a. This defines an embedding R H(R). Every relaion in he srucure R defines a corresponding relaion on H(R), and every funcion in he srucure R defines a funcion on H(R), hence he Hardy field H(R) can be endowed wih an S-srucure H(R). Given an L S -formula ϕ(x 1,..., x n ), and given definable funcions f 1,..., f n, we have H(R) ϕ([f 1 ],..., [f n ]) ε > 0 : (0, ε) : R ϕ(f 1 (),..., f n ()). See [7, Secion 5.3] for precise definiions and proofs. In paricular he S-srucure H(R) is an elemenary exension of he S-srucure R. Noe ha he operaions + and urn H(R) ino a field, he order urns i ino an ordered field, and also noe ha his field is real closed. Moreover, he S-srucure H(R) is o-minimal. Suppose ha S is an expansion of S, and ha R is an S -srucure expanding R. Then all funcions ha are definable in R are also definable in R. This defines an inclusion H(R) H(R ). Noe ha, by resricion, R has an S-srucure induced by his S -srucure. If ϕ(x 1,..., x n ) is an L S -formula, and h 1,..., h n H(R), hen H(R) ϕ(h 1,..., h n ) H(R ) ϕ(h 1,..., h n ).

17 Logarihmic limi ses of real semi-algebraic ses 17 In oher words he S-srucure on H(R ) is an elemenary exension of H(R). If R is polynomially bounded, for every definable funcion f whose germ is no 0, here exiss r in he field of exponens and c R \ {0} such ha f(x) lim x 0 + x r = c. If h is he germ of f, we denoe he exponen r by v(h). The map v : H(R) \ {0} R is a real valued valuaion, urning H(R) in a non-archimedean field of rank one. The image group of he valuaion is he field of exponens of R, denoed by Λ. The valuaion has a naural secion, he map Λ r x r H(R). The valuaion ring, denoed by O, is he se of all germs bounded in a neighborhood of zero, and he maximal ideal m of O is he se of all germs infiniesimal in zero. The valuaion ring O is convex wih respec o he order, hence he valuaion opology coincides wih he order opology. The map O R sending every elemen of O o is value in zero has kernel m, hence i idenifies in a naural way he residue field O/m wih R. We will usually denoe by H(R) he germ of he ideniy funcion. We have v() = 1. As an example we can describe he field H(R). Every elemen of his field is algebraic over he fracion field R(). Hence H(R) is he real closure of R(), wih reference o he unique order such ha > 0 and x R >0 : < x. The image of he valuaion is Q. Consider he real closed field of formal Puiseux series wih real coefficiens, R(( Q )) = n 1 R((1/n )). The elemens of his field have he form x r (s(x 1/n )), where r Z and s is a formal power series. We have R() R(( Q )) as ordered fields, hence H(R) R(( Q )). The elemens of H(R) are he elemens of R(( Q )) ha are algebraic over R(). For hese elemens he formal power series s is locally convergen. Anoher example is he field H(R an ), see [22]. By [22, Theorem B], for every elemen h of his field, here exiss a number r R, a formal power series F = α R 0 c α X α and a radius δ R >0 such ha c α R, {α c α 0} is well ordered, he series α c α δ α < +, (hence F is convergen and defines a coninuous funcion on [0, δ], analyic on (0, δ)) and h = [x r F (x)]. Le F be a real closed field exending R. The convex hull of R in F is a valuaion ring denoed by O. This valuaion ring defines a valuaion v : F Λ, where Λ is an ordered abelian group. We say ha F is a real closed non-archimedean field of rank one exending R if Λ has rank one as an ordered group, or, equivalenly, if Λ is isomorphic o an addiive subgroup of R. Hence real closed non-archimedean fields of rank one exending R have a real valued valuaion (non necessarily surjecive) well defined up o

18 18 Daniele Alessandrini a scaling facor. This valuaion is well defined when we choose an elemen F wih > 0 and v() > 0, and we choose a scaling facor such ha v() = 1. Now a valuaion v : F R is well defined, wih image v(f ) = Λ R. Consider he subfield R() F. The order induced by F has he propery ha > 0 and x R >0 : < x. Hence F conains he real closure of R() wih reference o his order, i.e. H(R). Moreover he valuaion v on F resrics o he valuaion we have defined on H(R), as, if O is he valuaion ring of F, O H(R) is precisely he valuaion ring O of H(R). In oher words every non-archimedean real closed field F of rank one exending R is a valued exension of H(R). 4.2 Non-Archimedean amoebas. Le F be a non-archimedean real closed field of rank one exending R, wih a fixed real valued valuaion v : F R. By convenion, we define v(0) =, an elemen greaer han any elemen of R. The map F h h = exp( v(h)) R 0 is a non-archimedean norm. The componen-wise logarihm map can be defined also on F by Log : (F >0 ) n (h 1,..., h n ) ( log( h 1 ),..., log( h n ) ) R n. Noe ha log( h ) = v(h). If V (F >0 ) n, he logarihmic image of V is he image Log(V ). Le S be a se of symbols expanding OS, and le (F, a) be an S-srucure expanding he OS-srucure on he non-archimedean real closed field F of rank one exending R. If V (F >0 ) n is a definable se in (F, a), we define he non-archimedean amoeba of V as he closure in R n of he logarihmic image of V, and we will wrie A(V ) = Log(V ). The case we are more ineresed in is when R = (R, a) is an o-minimal, polynomially bounded S-srucure expanding R, and H(R) is he Hardy field, wih is naural valuaion v and is naural S-srucure. Non-Archimedean amoebas of definable ses of H(R) are closely relaed wih logarihmic limi ses of definable ses of R. Le F K be wo real closed fields. Le S be a se of symbols expanding OS, le (F, a), (K, b) be S-srucures expanding he OS srucure on he real closed fields and such ha K is an elemenary exension of F. Le V F n be a definable se in (F, a). We will denoe by V K he exension of V o he srucure (K, b). For example, if V R n is a definable se in R, we can always define an exension V H(R) H(R) n of V o H(R). Lemma 4.1. Le R be an o-minimal polynomially bounded srucure. Le V (R >0 ) n be a definable se. Then (0,..., 0, 1) A 0 (V ) (0,..., 0, 1) Log ( V H(R)). Proof. Suppose ha (0,..., 0, 1) A ( V H(R)). Then here is a poin (x 1,..., x n ) V H(R) such ha v(x n ) = 1 and v(x i ) = 0 for all i < n. If f 1,..., f n are definable funcions such ha x i = [f i ], hen ε > 0 : (0, ε) : (f 1 (),..., f n ()) V. Moreover, when 0 we have ha f n () 0 and f i () a i > 0 for i < n. Hence

19 Logarihmic limi ses of real semi-algebraic ses 19 V conains a sequence ending o (a 1,..., a n 1, 0) wih a 1,... a n 1 0, and A 0 (V ) conains (0,..., 0, 1). Vice versa, suppose ha (0,..., 0, 1) A 0 (V ). Then, by Theorem 3.6 here exiss a sequence (x(k)) in V such ha (x(k)) (a 1,..., a n 1, 0), where a 1,..., a n 1 > 0. Le ε be a number less han all he numbers a 1,..., a n 1, and consider he se {x R x 1,..., x n 1 : x i a i < 1 2 ε and (x 1,..., x n 1, x) V }. As his se is definable, and as i conains a sequence converging o zero, i mus conain an inerval of he form (0, δ), wih δ > 0. In one formula: x (0, δ) : x 1,..., x n 1 : x i a i < 1 2 ε and (x 1,..., x n 1, x) V. This senence can be urned ino a firs order L S -formula using a definiion of V. This formula mus also hold for H(R). We can choose an x H(R), wih x > 0 and v(x) = 1. Then x < δ, hence x 1,..., x n 1 : x i a i < 1 2 ε and (x 1,..., x n 1, x) V H(R). Now v(x i ) = 0 for all i > 1, as x i a i < 1 2 ε. Hence Log(x 1,..., x n 1, x) = (0,..., 0, 1). Theorem 4.2. Le R be an o-minimal polynomially bounded srucure wih field of exponens Λ. Le V (R >0 ) n be a definable se. Then A 0 (V ) Λ n = Log ( V H(R)). In paricular, Log ( V H(R)) is a closed subse of Λ n. Proof. As Log ( V H(R)) Λ n, we only need o prove ha for all x Λ n, x A 0 (V ) x Log ( V H(R)). We choose a marix B wih enries in Λ sending x o (0,..., 0, 1). Then we conclude by he previous lemma applied o he definable se B(V ). For he las saemen, recall ha A 0 (V ) is a closed subse of R n, hence A 0 (V ) Λ n is a closed subse of Λ n. Theorem 4.3. Le F K be wo non-archimedean real closed fields of rank one exending R, wih a choice of a real valued valuaion defined by an elemen F. Denoe he value groups by Λ = v(f ) and Ω = v(k ). Le S be a se of symbols expanding OS, le (F, a), (K, b) be S-srucures expanding he OS srucure on he real closed fields and such ha K is an elemenary exension of F. Le V be a definable se in (F, a), and V K be is exension o (K, b). Then Log(V ) Λ n is dense in Log ( V K) Ω n. Proof. Suppose, by conradicion, ha x Log ( V K) and i is no in he closure of Log(V ). Then here exiss an ε > 0 such ha he cube C = {y R n y 1 x 1 < ε,..., y n x n < ε}

20 20 Daniele Alessandrini does no conain poins of Log(V ). Le h V K be an elemen such ha Log(h) = x, and le d F be an elemen such ha 0 < v(d) < ε. Consider he cube ( h1 ) ( E = d, h h2 ) ( 1d d, h hn ) 2d d, h nd K n. The image Log(E) is conained in C, hence E V is empy. Bu, as (K, b) is an elemenary exension of (F, a), also E V K is empy. This is a conradicion as h E and h V K. Corollary 4.4. Le S be a se of symbols expanding OS, and le R = (R, a) be an o- minimal polynomially bounded S-srucure wih field of exponens Λ, expanding R. Le V (R >0 ) n be a definable se in R. Suppose ha here exiss a subfield Ω R such ha Λ Ω and R Ω is o-minimal and polynomially bounded. Then A 0 (V ) Λ n is dense in A 0 (V ) Ω n. Proof. Consider he Hardy fields H(R) and H(R Ω ). By Theorem 4.2, A 0 (V ) Λ n = Log ( V H(R)) and A 0 (V ) Ω n = Log ( V ) H(RΩ ). As we said above, he S-srucures on H(R) and H(R Ω ) are elemenary equivalen. The saemen follows by he previous heorem. Corollary 4.5. Le S be a se of symbols expanding OS, and le R = (R, a) be a regular polynomially bounded S-srucure wih field of exponens Λ. Le V (R >0 ) n be a se ha is definable in R. We denoe by V H(R) he exension of V o H(R) and by V H(RR ) he exension of V o H(R R ). Then A 0 (V ) = Log ( V H(RR ) ). Moreover he subse A 0 (V ) Λ n is dense in A 0 (V ), and, as A 0 (V ) is closed, A ( V H(R)) = A ( V H(RR ) ) = Log ( V H(RR ) ). Corollary 4.6. Le V (R >0 ) n be a semi-algebraic se. Then A 0 (V ) Q n is dense in A 0 (V ). Le F be a non-archimedean real closed field of rank one exending R, and le V F be he exension of V o F. Then A 0 (V ) = A ( V F). If F exends H(R R ), hen A ( V F) = Log ( V F). As a furher corollary, we prove he following proposiion, ha will be needed laer. Proposiion 4.7. Le V (R >0 ) n be a se definable in a regular polynomially bounded srucure, and le π : R n R m be he projecion on he firs m coordinaes (wih m < n). Then we have A 0 (π(v )) = π(a 0 (V )).

21 Logarihmic limi ses of real semi-algebraic ses 21 Proof. We denoe by π H(R) : H(R) n H(R) m he projecion on he firs m coordinaes. The proposiion follows easily from Corollary 4.5 and from he fac ha π H(R)( V H(R)) = (π(v )) H(R). 4.3 Pachworking families. Le S be a se of symbols expanding OS, and le R = (R, a) be an S-srucure expanding R. Le F be a non-archimedean real closed field of rank one exending R, and le (F, b) be an S-srucure expanding he OS srucure on he real closed field and exending he S-srucure on R. If V is a definable se in (F, a), we have defined he non-archimedean amoeba A(V ) as he closure in R n of he logarihmic image Log(V ). Up o now, we sudied he properies of non-archimedean amoebas only in he paricular case in which V is he exension of a definable se in R, and his seems an essenial requiremen because we used exensively he properies of logarihmic limi ses o sudy non-archimedean amoebas. This paricular case can be called he consan coefficien case, because one can consider he elemens of F as real valued funcions (his inerpreaion is clear in he case of Hardy fields) and he elemens of R as consan real valued funcions, and he exension o F of a definable se in R is definable wih real coefficiens. The purpose of his subsecion is o show ha wih similar mehods one can sudy he properies of general definable ses. We will resric ourself o he case where (F, b) is he S-srucure H(R) on he Hardy field of R. The idea is o define a one parameer family of definable ses in R, and o show ha he logarihmic limi of his family is he non-archimedean amoeba. Families consruced wih his idea are someimes called pachworking families in he lieraure. Afer having shown ha he non-archimedean amoebas are polyhedral complexes wih conrolled dimension, we will show how heir local srucure around a poin λ is described by logarihmic limi ses of a paricular definable se in R ha we will call he iniial se of λ. If V (H(R) >0 ) n is a definable se, here exis a firs order L S -formula ϕ(x 1,..., x n, y 1,..., y m ), and parameers a 1,..., a m H(R) such ha V = {(x 1,..., x n ) ϕ(x 1,..., x n, a 1,..., a m )}. Choose definable funcions f 1,..., f m such ha [f i ] = a i. These daa define a definable se in R: Ṽ = { (x 1,..., x n, ) (R >0 ) n+1 ϕ(x 1,..., x n, f 1 (),..., f m ()) }. Suppose ha ϕ (x 1,..., x n, y 1,..., y m ) is anoher formula defining V wih parameers a 1,..., a m, and ha f 1,..., f m are definable funcions such ha [f i ] = a i. These daa define Ṽ = { (x 1,..., x n, ) (R >0 ) n+1 ϕ (x 1,..., x n, f 1(),..., f m ())}. As boh formulae define V we have H(R) x 1,..., x n : ϕ(x 1,..., x n, a 1,..., a m ) ϕ (x 1,..., x n, a 1,..., a m ).

22 22 Daniele Alessandrini As we said above, we have: ε > 0 : (0, ε) : R x 1,..., x n : ϕ(x 1,..., x n, f 1 (),..., f m ()) ϕ (x 1,..., x n, f 1(),..., f m ()). Hence Ṽ ( R n (0, ε) ) = Ṽ ( R n (0, ε) ), and he se Ṽ is well defined for small enough values of. Acually we prefer o see he se Ṽ as a paramerized family: V = { (x 1,..., x n ) (R >0 ) n (x 1,..., x n, ) Ṽ }. We can say ha he se V deermines he germ near zero of his paramerized family. We will use he noaion V = (V ) >0 for he family, and we will call hese families pachworking families deermined by V, as hey are a generalizaion of he pachworking families of [24]. Given a pachworking family V, we can define he ropical limi of he family as A 0 (V ) = lim 0 A (V ) = lim Log 1 0 (V ), where he limi is in he sense of subsecion 2.1. This is a closed subse of R n. Noe ha his se only depends on V. If V is he exension o H(R) of a definable subse W R n, hen he pachworking family V is consan: V = W, and he ropical limi is simply he logarihmic limi se: A 0 (V ) = A 0 (W ). Consider he logarihmic limi se of Ṽ : As in subsecion 3.2, we consider he se A 0 (Ṽ ) = lim 0 A (Ṽ ) Rn+1. H = { (x 1,..., x n ) R n (x 1,..., x n, 1) A 0 (Ṽ )}. Noe ha Log 1 (R n { 1}) = (R >0 ) n {}. Hence A 0 (V ) = lim 0 Log 1 (V ) = H. 1 Now consider he exension Ṽ H(R) of he Ṽ o he Hardy field H(R). By he resuls of he previous secion, we know ha A ( Ṽ H(R)) = A 0 (Ṽ ). If we denoe by H(R) he germ of he ideniy funcion, we have ha V = {(x 1,..., x n ) (x 1,..., x n, ) Ṽ H(R) }, as, for i {1,..., m}, we have f i () = a i. Hence, as log = 1, A(V ) H = A 0 (V ). Lemma 4.8. (0,..., 0) A 0 (V ) (0,..., 0) Log(V ). Proof. The forward implicaion follows from wha we said above. The converse follows from he second par of he proof of Lemma 4.1, applied o he se Ṽ.

23 Logarihmic limi ses of real semi-algebraic ses 23 Le λ Λ n. We define a wised se V λ = { x H(R) n ϕ( λ1 x 1,..., λn x n, a 1,..., a m ) }. Then λ Log(V ) (0,..., 0) Log(V λ ). Then we define H λ = { (x 1,..., x n ) R n (x 1,..., x n, 1) A 0 (Ṽ λ ) }. Now H λ is simply H ranslaed by he vecor λ. Hence we ge: Lemma 4.9. For all λ Λ, we have λ A 0 (V ) λ Log(V ). Using hese facs we can exend he resuls of he previous secions abou logarihmic limi ses and heir relaions wih non-archimedean amoebas o ropical limis of pachworking families. For example we can prove he following saemens. Theorem Le S be a srucure expanding OS, and le R = (R, a) be a regular polynomially bounded S-srucure wih field of exponens Λ. Le V H(R) be a definable se in he S-srucure H(R), and le V be a pachworking family deermined by V. Then he following facs hold: (1) A 0 (V ) is a polyhedral complex wih dimension less han or equal o he dimension of V. (2) A 0 (V ) Λ n = Log(V ). (3) A 0 (V ) = A(V ). (4) A 0 (V ) Λ n is dense in A 0 (V ). Proof. Every saemen follows from he corresponding saemen abou logarihmic limi ses, and from he facs exposed above. V λ For every poin λ Λ, he wised se V λ defines a germ of pachworking family = (V λ ) >0, where for every, V λ is a definable subse of (R >0 ) n. Consider he se V0 λ = lim V λ, 0 where he limi is defined as in subsecion 2.1. The se V0 λ is a definable subse of (R >0) n, and is logarihmic limi se A 0 (V0 λ ) can be used o describe a neighborhood of λ in A(V ), as we will sae in he nex heorem. The proof follows from Theorem 3.10 applied o he definable se Ṽ λ. The se V0 λ, in he language of Theorem 3.10, is called W. Theorem Le S be a se of symbols expanding OS, and le R = (R, a) be an S-srucure expanding R ha is o-minimal and polynomially bounded, wih field of exponens Λ. Le V H(R) be a definable se in he S-srucure H(R). Then we have λ Λ n : λ A(V ) V λ 0. Moreover, if Λ = R, for all λ R, here exiss a neighborhood U of λ in A(V ) such ha he ranslaion of U by λ is a neighborhood of (0,..., 0) in A 0 (V λ 0 ).

24 24 Daniele Alessandrini Proof. I follows from he argumens above and from Theorem The se V0 λ is well defined for all λs, and i depends only on λ. I can be called he iniial se of λ, as i plays he role of he iniial ideal of [17]. The difference is ha V0 λ is a geomeric objec, while he iniial ideal of [17] is an algebraic objec. 5 Comparison wih oher consrucions 5.1 Complex algebraic ses. Logarihmic limi ses of complex algebraic ses are a paricular case of logarihmic limi ses of real semi-algebraic ses, in he following sense. Le V C n be a complex algebraic se, and consider he real semi-algebraic se V = { x (R >0 ) n z V : z = x }. The logarihmic limi se of V as defined in [3] is precisely he logarihmic limi se of V in our noaion. We will wrie A 0 (V ) = A 0 ( V ). Hence all he resuls we go abou logarihmic limi ses of real semi-algebraic ses produce an alernaive proof of he same resuls for complex algebraic ses, ha were originally proved parly in [3] and [4]. Even he descripion of logarihmic limi ses via non-archimedean amoebas can be ranslaed o complex algebraic ses. Le F be a non-archimedean real closed field of rank one exending R, and le v be a choice of a real valued valuaion on F, as in subsecion 4.1. The field K = F[i] is an algebraically closed field exending C, wih an exended valuaion v : K R defined by v(a + bi) = min(v(a), v(b)). The componen-wise logarihm map can be exended o he field K by Log : K n (z 1,..., z n ) ( v(z 1 ),..., v(z n )) R n. On K here is also he complex norm : K F 0 defined by a + bi = a 2 + b 2. Now if V is an algebraic se in K n, he se V = { x (F >0 ) n z V : z = x } is a semi-algebraic se in F n. The logarihmic image of V is defined as he image Log(V ), and he non-archimedean amoeba A(V ) is defined as he closure of his image. Clearly, we have Log(V ) = Log( V ) and A(V ) = A( V ). Moreover, if V C n is an algebraic se, and V K K n is is exension o K, hen ( V ) K = V K. These facs direcly give he relaion beween logarihmic limi ses of complex algebraic ses and non-archimedean amoebas in algebraically closed fields. The same relaion holds wih pachworking families. Le R = (R, a) be a regular polynomially bounded srucure wih field of exponens Λ, le F = H(R) and K = H(R)[i] and le V K n be an algebraic se. There are polynomials f 1,..., f m K[x 1,..., x n ] such ha V = V (f 1,..., f m ). Every polynomial f j has he form f j = ω Z n (a j,ω + ib j,ω )x ω,

25 Logarihmic limi ses of real semi-algebraic ses 25 where a j,ω, b j,ω H(R). Choose represenaives funcions α j,ω, β j,ω such ha [α j,ω ] = a j,ω, [β j,ω ] = b j,ω. This choice defines families of polynomials f j, = ω S(f) and a corresponding family of algebraic ses in C n (α j,ω () + iβ j,ω ())x ω V = V (f 1,,..., f m, ). We will call hese families pachworking families because hey generalize he pachworking polynomial of [11, Par 2], and we will denoe he family by V = (V ). The family V depends of he choice of he polynomials f j and of he definable funcions α j,ω, β j,ω. If we change hese choices we ge anoher pachworking family coinciding wih V for (0, ε). The ropical limi of one such family is A 0 (V ) = lim 0 A (V ) where he limi is defined as in subsecion 2.1. As before, V is a semi-algebraic se in H(R) n, and if V = ( V ) is a pachworking family defined by V, hen here exiss an ε > 0 such ha for (0, ε) we have V = V. Hence A 0 (V ) = A 0 ( V ), and we can ge he properies of he ropical limi of complex pachworking families as a corollary of he properies of ropical limis of real pachworking families. Le f R[x 1,..., x n ]. Le V be he inersecion of he zero locus of f and (R >0 ) n, and le V C be he zero locus of f in C n. As V V C, he logarihmic limi se of V is included in he logarihmic limi se of V C. Moreover, as V C is a complex hypersurface, i is possible o give an easy combinaorial descripion of A 0 (V C ), i is simply he dual fan of he Newon polyope of f. Unforunaely, i is no possible, in general, o use his fac o undersand he combinaorics of A 0 (V ). There are examples where V is an irreducible hypersurface, and A 0 (V ) is a subpolyhedron of A 0 (V C ) ha is no a subcomplex. For example, consider he real algebraic se W in Figure 6; he logarihmic limi se of W is only he ray in he direcion ( 1, 0, 0), bu his ray lies in he inerior of a face of A 0 (W C ), for he deails see he end of he nex subsecion. 5.2 Posiive ropical varieies. In his subsecion we compare he noion of non-archimedean amoebas for real closed fields ha we sudied in his paper wih a similar objec called posiive ropical variey sudied in [18]. To be consisen wih [18], we will denoe by K = n=1 C((1/n )) he algebraically closed field of formal Puiseux series wih complex coefficiens, whose se of exponens is an arihmeic progression of raional numbers, and by F = n=1 R((1/n )) he subfield of series wih real coefficiens. K is he algebraic closure of F. These fields have a naural valuaion v : K Q, wih valuaion ring O, and residue map r : O C. Noe ha he valuaion v is compaible wih he order of F, i.e. he valuaion ring O F is convex for he order, and ha r(o F) = R.

26 26 Daniele Alessandrini Figure 6. W = {(x, y, z) R 3 x 2 (1 (z 2) 2 ) = x 4 + (y 1) 2 } is an irreducible surface, bu i has a sick, he line {y = 1, x = 0}. The logarihmic limi se of W (R >0 ) 3 is only he ray in he direcion ( 1, 0, 0), bu his ray is conained in he inerior par of a face of he dual fan of he Newon polyope of he defining polynomial x 2 (1 (z 2) 2 ) x 4 (y 1) 2. We will denoe by F >0 he se of posiive elemens of he field F. Following [18] we will also use he noaion: Le V be an algebraic se in K n. The se F + = {z K r ( z v(z)) R >0 }. V >0 = V (F >0 ) n is a semi-algebraic se, whose non-archimedean amoeba A(V >0 ), i.e. he closure of he logarihmic image Log(V >0 ), has been sudied in subsecion 4.3. In [18] a similar definiion is given. The posiive par of V is V + = V (F + ) n. The closure of Log(V + ) is called posiive ropical variey, and i is denoed by Trop + (V ). From he definiion i is clear ha A(V >0 ) Trop + (V ). In many examples he ses A(V >0 ) and Trop + (V ) coincide, bu i is also possible o consruc examples where he inclusion is sric. For example V = { (x 1, x 2 ) K 2 x (x 2 1) 2 x 3 1 = 0 }. Then V >0 is he exension o F of he se in Figure 7, and A(V >0 ) = {(x 1, x 2 ) R 2 x 1 = 0, x 2 0 or x 1 0, 2x 2 = 3x 1 } Trop + (V ) = A(V >0 ) {x 2 = 0, x 1 0}. A more ineresing example where A(V >0 ) Trop + (V ) is he following: consider he se W = { (x, y, z) R 3 x 2 (1 (z 2) 2 ) = x 4 + (y 1) 2},