aa r X i v : . [ m a t h . L O ] J u l Differentiability Properties ofLog-Analytic Functions
Tobias Kaiser and Andre Opris
Abstract.
We show that the derivative of a log-analytic function is log-analytic. We provethat log-analytic functions exhibit strong quasianalytic properties. We establish the para-metric version of Tamm’s theorem for log-analytic functions.
Introduction
Log-analytic functions have been defined by Lion and Rolin in their seminalpaper [8]. Their definition is kind of hybrid. They are iterated compositionsfrom either side of globally subanalytic functions (i.e. functions definable inthe o-minimal structure R an of restricted analytic functions) and the globallogarithm. An interesting class is formed by the constructible functions whereroughly globally subanalytic functions are composed with the logarithm on-ce from the left. By Cluckers and Dan Miller [1], the class of constructiblefunctions is stable under parametric integration. From the viewpoint of logic,log-analytic functions are definable in the o-minimal expansion R an , exp of R an by the global exponential function; in fact they generate the whole structure R an , exp . But from the point of analysis their definition avoids the exponenti-al function and should therefore also not exhibit properties of the functionexp( − /x ) as flatness or infinite differentiability but not real analyticity. Thisseems to be obvious. But the problem is that a composition of globally sub-analytic functions and the logarithm allows a representation by ’nice’ termsonly piecewise. Moreover the ’pieces’ are in general not definable in R an butonly in R an , exp as the following example shows (where the global logarithm isextended by 0 to the real line). Example
Let F ( x, y ) be the globally subanalytic function with F ( x, y ) = y − x if y >x and zero else. Let f ( x, y ) be the log-analytic function F ( x, log( y )). Then f ( x, y ) = log( y ) − x if y > exp( x ) and zero else.If a log-analytic function has on some ’piece’ a representation by nice termsthen on this piece the function is real analytic. But it is much harder to controlthe function on the boundary of the piece as the following example indicates. Keywords and phrases: log-analytic functions, preparation, differentiability, Tamm’s theorem xample Let f ( x, y ) be the log-analytic function with f ( x, y ) = y − x/ log( y ) if y > x = 0 we have f ( x, − ) ∼ − / log( y ) and f (0 , − ) ∼ y as y ց f ( x, y ) where y is the last va-riable can be piecewise written as f ( x, y ) = a ( x ) y ( x, y ) q · · · y r ( x, y ) q r u ( x, y )where y ( x, y ) = y − Θ ( x ) , y ( x, y ) = log( | y ( x, y ) | ) − Θ ( x ) , . . . , the q j ’s arerational exponents and u ( x, y ) is a unit of a special form. This gives roughlythat the functions f ( x, − ) behave piecewise as iterated logarithms indepen-dently of x where the order of iteration is bounded in terms of f . But theproblem is that the functions a ( x ) , Θ ( x ) , . . . , Θ r ( x ) although being definablein R an , exp are in general not log-analytic anymore. We will present an examplebelow. To be able to use the preparation theorems our initial result is the keyobservation that on certain ’pieces’ which we call simple a log-analytic functi-on can be prepared with log-analytic data only. For this one has do redo theproof of the existing preparation result. This allows us to prove that the classof log-analytic functions ist closed under taking derivatives. Theorem A
Let U ⊂ R n be open and let f : U → R be log-analytic. Let i ∈ { , . . . , m } be such that f is differentiable with respect to x i on U . Then ∂f /∂x i is log-analytic. The asymptotic behaviour of log-analytic functions given by the preparartionresults on simple sets implies strong quasianalyticity (see Miller [9] for thisresult in polynomially bounded o-minimal structures).
Theorem B
Let U ⊂ R n be a domain and let f : U → R be a log-analytic function. Thenthere is N ∈ N with the following property. If f is C N and if there is a ∈ U suchthat all derivaties of f up to order N vanish in a then f vanishes identically. With the results above, we can generalize the parametric version of Tamm’stheorem for globally subanalytic functions by Van den Dries and Miller [4] (seeTamm [10] for the original version) to log-analytic functions by adapting theirarguments. 2 heorem C
Let f : R n × R m → R , ( x, y ) f ( x, y ) , be a log-analytic function. Then thereis N ∈ N such that the following holds for every ( x, y ) ∈ R n × R m . If f ( x, − ) is C N at y then f ( x, − ) is real analytic at y . This implies in particular that the set of all x ∈ R n such that f ( x, − ) is realanalytic is definable in R an , exp and therefore o-minimal. This shows again thatthe class of log-analytic functions is a proper extension of the class of globallysubanalytic functions but shares its properties from the viewpoint of analysisand of o-minimality. The latter observation does not hold when the exponentialfunction comes into the game as remarked at the end of [4]. We formulate itin the following way: Example
Consider the function f : R × R → R , ( x, y ) | y | | x | , y = 0 , if0 , y = 0 . which is definable in R exp . Then the set of all x ∈ R such that f ( x, − ) isreal-analytic is the set of integers.The paper is organized as follows: In Section 1 we give the definition of log-analytic functions and formulate elementary properties. In Section 2 we presentthe preparation theorem of Lion-Rolin for log-analytic functions and prove thepure preparation theorem on simple sets. Section 3 is devoted to the proof ofthe above theorems. Notations:
The empty sum is by definition 0 and the empty product is by definition 1.By N = { , , . . . } we denote the set of natural numbers and by N = { , , , . . . } the set of nonnegative integers. Given x ∈ R let ⌈ x ⌉ be the smallest integerwhich is not smaller than x .We set R > := { x ∈ R | x > } . For m, n ∈ N we denote by M ( m × n, R ) theset of m × n -matrices with real entries. For P ∈ M ( m × n, R ) we denote by t P ∈ M ( n × m, R ) its transpose. Given x ∈ R \{ } we denote by sign( x ) ∈ {± } its sign. For a, b ∈ R with a ≤ b we denote by [ a, b ] the closed interval andby ] a, b [ the open interval with endpoints a, b , respectively. Denoting by | | theeuclidean norm on R n we set S n − := { x ∈ R n | | x | = 1 } and, for a ∈ R n and ε ∈ R > , B ( a, ε ) := { x ∈ R n | | x − a | < ε } . Given a subset A of R n we denoteby A its closure. A domain in R n is a nonempty, open and connected subsetof R n . 3e use the usual o -notation and O -notation. By the symbol ∼ we denoteasymptotic eqivalence. By log k we denote the k -times iterated of the logarithm.We assume familarity with basic facts about o-minimality as can be found inVan den Dries [2] and basic knowledge of the o-minimal structures R an and R an , exp as can be found in Van den Dries and Miller [5]. Conventions:
From now on definable means definable in the o-minimal structure R an , exp withparameters. A definable cell is assumed to be analytic. We give the definition of a logarithmic-analytic function (log-analytic for short)established by Lion and Rolin in [8].
Let X ⊂ R m be definable and let f : X → R be a function.(a) Let r ∈ N . By induction on r we define that f is log-analytic of orderat most r . Base case:
The function f is log-analytic of order at most 0 if f ispiecewise the restriction of globally subanalytic functions; i.e. there is afinite decomposition C of X into definable sets such that for C ∈ C thereis a globally subanalytic function F : R m → R such that f | C = F | C . Inductive step:
The function f is log-analytic of order at most r if thefollowing holds. There is a finite decomposition C of X into definable setssuch that for C ∈ C there are k, l ∈ N , a globally subanalytic function F : R k + l → R and log-analytic functions g , ..., g k : C → R , h , . . . , h l : C → R > of order at most r − f | C = F (cid:0) g , ..., g k , log( h ) , ..., log( h l ) (cid:1) . (b) Let r ∈ N . We call f log-analytic of order r if f is log-analytic oforder at most r but not of order at most r − f log-analytic if it is log-analytic of order r for some r ∈ N .4 .2 Remark (1) A log-analytic function is definable.(2) Let L an be the natural language for R an ; i.e. L an is the augmentationof the language L of ordered rings by symbols for restricted analyticfunctions. Then the log-analytic functions are precisely those definablefunctions which are piecewise given by L an ( − , ( n √ ... ) n =2 , ,... , log)-terms(compare with Van den Dries et al. [3]).(3) A function is log-analytic of order 0 if and only if it is piecewise therestriction of globally subanalytic functions. Let X ⊂ R n be definable.(1) Let r ∈ N . The set of log-analytic functions on X of order at most r isa ring with respect to pointwise addition and multiplication.(2) The set of log-analytic functions on X is a ring with respect to pointwiseaddition and multiplication. We let x = ( x , . . . , x n ) range over R n and y over R . We set π : R n × R → R n , ( x, y ) x . Lion and Rolin [8] have established a preparation result for log-analytic func-tions which we state here.Let r ∈ N . We let w = ( w , . . . , w r ) range over R r +1 . We set π ∗ : R n × R r +1 → R n , ( x, w ) x . Let r ∈ N and let D ⊂ R n × R r +1 be definable. A function u : D → R iscalled a special unit on D if u = v ◦ ϕ where the following holds:(a) The function ϕ is given by ϕ : D → [ − , s ,ϕ ( x, w ) = (cid:16) b ( x ) r Y j =0 | w j | p j , . . . , b s ( x ) r Y j =0 | w j | p sj (cid:17) , where s ∈ N , p ij ∈ Q for ( i, j ) ∈ { , ..., s } × { , ..., r } and b , ..., b s aredefinable functions on π ∗ ( D ) which have no zeros.5b) The function v is a real power series which converges absolutely on anopen neighbourhood of [ − , s .(c) There are d , d ∈ R > such that d ≤ v ≤ d on [ − , s .We call b := ( b , ..., b s ) a tuple of base functions for u and I := (cid:0) s, v, b, P (cid:1) where P := p · · p r · ·· · p s · · p sr ∈ M (cid:0) s × ( r + 1) , R (cid:1) a describing tuple for u .Let C ⊂ R n × R be definable and let r ∈ N . A tuple Y = ( y , ..., y r ) of functions on C is called an r -logarithmic scale on C with center Θ = (Θ , . . . , Θ r ) if the following holds:(a) For every j ∈ { , . . . , r } we have y j > y j < j are definable functions on π ( C ) for every j ∈ { , . . . , r } .(c) It is y ( x, y ) = y − Θ ( x ) and y j ( x, y ) = log( | y j − ( x, y ) | ) − Θ j ( x ) for j ∈ { , . . . , r } .(d) There is ε ∈ ]0 ,
1[ such that 0 < | y ( x, y ) | < ε | y | for all ( x, y ) ∈ C or Θ = 0, and for every j ∈ { , ..., r } there is ε j ∈ ]0 ,
1[ such that0 < | y j ( x, y ) | < ε j | log( | y j − ( x, y ) | ) | for all ( x, y ) ∈ C or Θ j = 0.Note that Θ is uniquely determined by Y . Note also that Y is log-analytic ifand only if Θ is log-analytic. Let Y = ( y , . . . , y r ) be an r -logarithmic scale on C . Its sign sign( Y ) ∈{− , } r +1 is defined bysign( Y ) = (cid:0) sign( y ) , . . . , sign( y r ) (cid:1) . Let Y = ( y , . . . , y r ) be an r -logarithmic scale on C . Let q = ( q , . . . , q r ) ∈ Q r +1 . We set |Y | ⊗ q := r Y j =0 | y j | q j . .5 Definition Let Y be an r -logarithmic scale on C . We set C Y := (cid:8)(cid:0) x, Y ( x, y ) (cid:1) (cid:12)(cid:12) ( x, y ) ∈ C (cid:9) ⊂ R n × R r +1 . Let g : C → R be a function. We say that g is r -log-analytically preparedwith respect to y if g ( x, y ) = a ( x ) |Y ( x, y ) | ⊗ q u (cid:0) x, Y ( x, y ) (cid:1) where a is a definable function on π ( C ) which vanishes identically or has nozero, Y is an r -logarithmic scale on C , q ∈ Q r +1 and u is a special unit on C Y .The function a is called coefficient of g and base functions b , ..., b s of u arecalled base functions of g . We call J := (cid:0) r, Y , a, q, I (cid:1) where I is a describingtuple for u an r -preparing tuple for g . Let g : C → R be a function. If g is r -log-analytically prepared with respectto y then g is definable but not necessarily log-analytic. [8, Section 2.1] Let X ⊂ R n × R be definable and let f : X → R be a log-analytic function oforder r . Then there is a definable cell decomposition C of X such that f | C is r -log-analytically prepared with respect to y for every C ∈ C . Here is the promised example that the above can in general not be carried outin the log-analytic category.
Let ϕ : ]0 , + ∞ [ → R , t t/ (1 + t ). Consider the log-analytic function f : R > × R , ( x, y )
7→ − ϕ ( y )) − x. Then f is log-analytic of order 1 but does not allow a 1-log-analytic preparationwith log-analytic data only. Proof:
Assume that the contrary holds. Let C be a corresponding cell decomposition.Set ψ : ]0 , → R , t t/ (1 − t ). Then ψ is the compositional inverse of ϕ . Notethat f ( x, ψ ( e − /x )) = 0 for all x ∈ R > . Let α : R > → R , x ψ ( e − /x ). Then7 is not log-analytic and α ( x ) = P ∞ n =1 e − n/x for all x ∈ R > . By passing to afiner cell decomposition we find a cell C of the form C := n ( x, y ) ∈ R > × R > (cid:12)(cid:12) < x < ε, α ( x ) < y < α ( x ) + η ( x ) o with some suitable ε ∈ R > and some definable function η : ]0 , ε [ → R > suchthat f is 1-log-analytically prepared on C with log-analytic data only. Let (cid:0) r, Y , a, q, I (cid:1) be a 1-preparing tuple for f | C and let Θ = (Θ , Θ ) be the centerof Y . Claim: Θ = 0. Proof of the claim:
Assume that Θ is not the zero function. By the definitionof a 1-logarithmic scale we find ε ∈ ]0 ,
1[ such that | y | < ε | y | on C . Thisimplies α ( x ) ≤ | α ( x ) − θ ( x ) | for all 0 < x < ε . But this is not possible sincewe have α = o (Θ ) at 0 by the assumption that Θ is log-analytic and not thezero function. (cid:4) Claim
From f ( x, y ) = a ( x ) |Y ( x, y ) | ⊗ q u ( x, Y ( x, y ))for all ( x, y ) ∈ C and lim y ց α ( x ) f ( x, y ) = 0 for all x ∈ ]0 , ε [ we get by o-minimality that there is, after shrinking ε > j ∈ { , } such that lim y ց α ( x ) | y j ( x, y ) | q j = 0 for all x ∈ ]0 , ε [. By the claim the case j = 0is not possible. In the case j = 1 we have, again by the claim, that q > = log( α ). But this is a contradiction to the assumption thatthe function Θ is log-analytic since the function log( α ) on the right is notlog-analytic. This can be seen by applying the logarithmic series. We obtainthat log( α ( x )) + 1 /x ∼ e − /x . (cid:4) In the case of globally subanalytic functions (i.e. log-analytic functions of order0) there is the following well known stronger result. [8, Section 1]If f : X → R is globally subanalytic then there is a globally subanalytic celldecomposition C of X such that f | C is globally subanalytic prepared withrespect to y for all C ∈ C ; i.e. f | C ( x, y ) = a ( x ) | y − Θ( x ) | q v (cid:0) b ( x ) | y − Θ( x ) | p , . . . , b s ( x ) | y − Θ( x ) | p s (cid:1) where a, b , . . . , b s , Θ : π ( C ) → R are globally subanalytic functions such thatthe image of (cid:0) b ( x ) | y − Θ( x ) | p , . . . , b s ( x ) | y − Θ( x ) | p s (cid:1) is contained in [ − , s and v is a real power series converging absolutely on [ − , s with v ([ − , s ) ⊂ R > . 8 .2 Simple sets and simple preparation Let C ⊂ R n × R be definable and r ∈ N . (a) We call C r -admissible if there is an r -logarithmic scale on C .(b) We call C r -unique if there is a unique r -logarithmic scale on C . An r -logarithmic scale on C is called elementary if its center is vanishing. An elementary r -logarithmic scale may not exist on C . If it exists it is uniquelydetermined and log-analytic. If C has an elementary r -logarithmic scale then we call C r -elementary . Theelementary r -logarithmic scale on C is then denoted by Y el r = Y el r,C .For the next definition compare with the setting of [4, Section 4] and [7, Defi-nition 3.6]. Given x ∈ R n , we set C x := { y ∈ R | ( x, y ) ∈ C } . We call C simple if for every x ∈ π ( C ) we have C x = ]0 , d x [ for some d x ∈ R > ∪ { + ∞} . Let C be a definable cell decomposition of R n × R > . Then R n = [ { π ( C ) | C ∈ C simple } . We set e := 0 and e r := exp( e r − ) for r ∈ N . In the following we let 1 / ∞ . Let C be simple and r -elementary and let Y el r,C = ( y , . . . , y r ) . Then the follo-wing holds:(1) sup C x ≤ /e r for all x ∈ π ( C ) .(2) y = y, y = log( y ) , y j = log j − ( − log( y )) for j ∈ { , . . . , r } . sign( Y el r,C ) = (1 , − , , . . . , ∈ R r +1 . Proof:
For x ∈ π ( C ) let d x := sup C x . Let ( σ , . . . , σ r ) be the sign of Y el r,C . We show byinduction on k ∈ { , . . . , r } that d x ≤ /e k for all x ∈ π ( C ), that y = y, y =log( y ) , y j = log j − ( − log( y )) for all j ∈ { , . . . , k } and that ( σ , . . . , σ k ) =(1 , − , , . . . , ∈ R k +1 . k = 0: We have y = y by Definition 2.2. This gives σ = 1. That d x ≤ + ∞ =1 /e for all x ∈ π ( C ) is clear. k = 1: Since y = y and σ = 1 by above we obtain according to Definition 2.2that y = log( y ) and that d x ≤ /e for all x ∈ π ( C ). This gives σ = − k = 2: Since σ = − y <
0. According to Definition 2.2 we get that y = log( − y ) = log( − log( y )) and therefore that d x ≤ / exp(1) = 1 /e and σ = 1. k → k + 1: We can assume that k ≥
2. By the inductive hypothesis we have y k = log k − ( − log( y )) > σ k = 1. According to Definition 2.2 we obtainthat y k +1 = log k ( − log( y )) and that d x ≤ / exp( e k ) = 1 /e k +1 for all x ∈ π ( C ).This gives also σ k +1 = 1. (cid:4) We call
C r -simple if it is simple and r -admissible. Let C be r -simple. Then C is r -elementary and r -unique. Proof:
Let Y = ( y , . . . , y r ) be an r -logarithmic scale on C . We show that Y is ele-mentary and are done by Remark 2.13. Let Θ = (Θ , . . . , Θ r ) be the center of Y . We show by induction on k ∈ { , . . . , r } that Θ = . . . = Θ k = 0. k = 0: Assume that Θ = 0. Then by Definition 2.2 there is ε ∈ ]0 ,
1[ suchthat | y − Θ ( x ) | < ε | y | for all ( x, y ) ∈ C . Let x ∈ π ( C ) such that Θ ( x ) = 0. Then we obtain+ ∞ = lim y ց (cid:12)(cid:12)(cid:12) − Θ ( x ) y (cid:12)(cid:12)(cid:12) ≤ ε which is a contradiction. k = 1: Assume that Θ = 0. By the case k = 0 and Proposition 2.17 we have y = y . According to Definition 2.2 there is ε ∈ ]0 ,
1[ such that | log( y ) − Θ ( x ) | < ε | log( y ) | x, y ) ∈ C . Therefore1 = lim y ց (cid:12)(cid:12)(cid:12) − Θ ( x )log( y ) (cid:12)(cid:12)(cid:12) ≤ ε for x ∈ π ( C ) which is a contradiction. k = 2: Assume that Θ = 0. By the case k = 1 and Proposition 2.17 we have y = log( y ). According to Definition 2.2 there is ε ∈ ]0 ,
1[ such that | log( − y ) − Θ ( x ) | < ε | log( − y ) | for all ( x, y ) ∈ C . Therefore1 = lim y ց (cid:12)(cid:12)(cid:12) − Θ ( x )log( − y ) (cid:12)(cid:12)(cid:12) ≤ ε for x ∈ π ( C ) which is a contradiction. k → k +1: We may assume that k ≥
2. Assume that Θ k +1 = 0. By the inductivehypothesis and Proposition 2.16 we have y k = log k − ( − log( y )). According toDefinition 2.2 there is ε k +1 ∈ ]0 ,
1[ such that | log( y k ) − Θ k +1 ( x ) | < ε k +1 | log( y k ) | for all ( x, y ) ∈ C . Therefore1 = lim y ց (cid:12)(cid:12)(cid:12) − Θ k +1 ( x )log( y k ) (cid:12)(cid:12)(cid:12) < ε k +1 for x ∈ π ( C ) which is a contradiction. (cid:4) Let C be simple. Then the following are equivalent:(i) C is r -simple.(ii) sup C x ≤ /e r for every x ∈ π ( C ) . Proof: (i) ⇒ (ii): If C is r -simple then C is r -elementary by Proposition 2.19. ByProposition 2.17 we obtain (ii).(ii) ⇒ (i): Let y = y, y := log( y ) , y j = log j − ( − log( y )) for j ∈ { , . . . , r } .Then it is straightforward to see that Y := ( y , . . . , y r ) is a well defined ele-mentary r -logarithmic scale on C . (cid:4) .21 Definition Let g : C → R be a function. We say that g is elementary r -log-analyticallyprepared with respect to y if g is r -log-analytically prepared with elemen-tary r -logarithmic scale. Let C be simple and let g : C → R be r -log-analytically prepared with respectto y . Then g is elementary r -log-analytically prepared with respect to y . Proof:
Since C is simple and since g has an r -log-analytic preparation we have that C is r -simple. We are done by Proposition 2.19. (cid:4) Let q = ( q , . . . , q r ) ∈ Q r +1 with q = 0. We set j ( q ) := min { j | q j = 0 } and σ ( q ) := sign( q j ( q ) ) ∈ {± } . Moreover, let q diff := (cid:0) q − , . . . , q j ( q ) − , q j ( q )+1 , . . . , q r (cid:1) . Let C be r -simple. Let q ∈ Q r +1 with q = 0. Thenlim y ց |Y el r,C ( y ) | ⊗ q = , j ( q ) = 0 , σ ( q ) = +1 , + ∞ , j ( q ) = 0 , σ ( q ) = − , + ∞ , j ( q ) > , σ ( q ) = +1 , , j ( q ) > , σ ( q ) = − , Let C be r -simple. Let q ∈ Q r +1 with q = 0 . Then lim y ց (cid:12)(cid:12)(cid:12) ddy |Y el r,C ( y ) | ⊗ q |Y el r,C ( y ) | ⊗ q diff (cid:12)(cid:12)(cid:12) = q j ( q ) . Proof:
Let Y el r,C = ( y , . . . , y r ). We get by Proposition 2.17 that d | y | /dy = 1 and thatfor j ∈ { , . . . , r } d | y j | dy = − Q j − i =0 | y i | . ddy r Y j =0 | y j | q j = q | y | q − | y | q · . . . · | y r | q r − r X j =1 q j | y j | q j − Q j − i =0 | y i | Y i = j | y i | q i = q | y | q − Y i> | y i | q i − r X j =1 q j Y i ≤ j | y i | q i − Y i>j | y i | q i . We obtain the assertion by the growth properties of the iterated logarithms. (cid:4)
Let C ⊂ R n × R be definable. Let g : C → R be a function. We say that g is purely r -log-analyticallyprepared with respect to y if g is r -log-analytically prepared with respect to y with log-analytic logarithmic scale, log-analytic coefficient and log-analyticbase functions. An r -preparing tuple for g with log-analytic components is thencalled a purely r -preparing tuple for g . Let g : C → R be a function. If g is purely r -log-analytically prepared withrespect to y then g is log-analytic.In the next proposition and corollary we assume that r ≥ Let C be r -simple.(1) Let g : C → R be purely ( r − -log-analytically prepared with respect to y . Then there are t ∈ N , a log-analytic function η : π ( C ) → R t and aglobally subanalytic function G : R t × R r → R such that g ( x, y ) = G ( η ( x ) , Y el r − ,C ( y )) for all ( x, y ) ∈ R n × R .(2) Let h : C → R > be purely ( r − -log-analytically prepared with respectto y . Then there are t ∈ N , a log-analytic function η : π ( C ) → R t and aglobally subanalytic function H : R t × R r +1 such that log( h ( x, y )) = H ( η ( x ) , Y el r,C ( y )) for all ( x, y ) ∈ R n × R . roof: (1): By Corollary 2.20 we have that C is ( r − r − J for g of the form J = (cid:0) r − , Y el r − ,C , a, q, s, v, b, P (cid:1) where a, b , . . . , b s are log-analytic functions on π ( C ). Take t = s + 1 and η = ( η , . . . , η s ) : π ( C ) → R t , x (cid:0) a ( x ) , b ( x ) , . . . , b s ( x ) (cid:1) . Then η is log-analytic. Let z = ( z , . . . , z s ) and w = ( w , . . . , w r − ). Set α : R t × R r → R , ( z, w ) z r − Y j =0 | w j | q j . For i ∈ { , . . . , s } let α i : R t × R r → R , ( z, w ) z i r − Y j =0 | w j | p ij . Set G : R t × R r , ( z , . . . , z s , w , . . . , w r − ) ( α ( z, w ) v (cid:16) α ( z, w ) , . . . , α s ( z, w ) (cid:17) , | α i ( z, w ) | ≤ i ∈ { , . . . , s } , , else . Then G is globally subanalytic and we have g ( x, y ) = G ( η ( x ) , Y el r − ,C ( y ))for all ( x, y ) ∈ C .(2): By Corollary 2.20 we have that C is ( r − r − J for h of the form J = (cid:0) r − , Y el r − ,C , a, q, s, v, b, P (cid:1) where a, b , . . . , b s are log-analytic functions on π ( C ). Then a >
0. Take t = s + 1 and η = ( η , . . . , η s ) : π ( C ) → R t , x (cid:0) log( a ( x )) , b ( x ) , . . . , b s ( x ) (cid:1) . Then η is log-analytic. Let z = ( z , . . . , z s ) and w = ( w , . . . , w r ). Set β : R t × R r → R , ( z, w ) z + r − X j =0 q j w j +1 . i ∈ { , . . . , s } let α i : R t × R r → R , ( z, w ) z i r − Y j =0 | w j | p ij . Set H : R t × R r , ( z , . . . , z s , w , . . . , w r − ) ( β ( z, w ) + log (cid:16) v (cid:16) α ( z, w ) , . . . , α s ( z, w ) (cid:17)(cid:17) , | α i ( z, w ) | ≤ i ∈ { , . . . , s } , , else.Then H is globally subanalytic since log( v ) is globally subanalytic and we havelog( h ( x, y )) = H ( η ( x ) , Y el r,C ( y ))for all ( x, y ) ∈ C . (cid:4) Let C be r -simple and let g , . . . , g k : C → R and h , . . . , h l : C → R > be purely ( r − -log-analytically prepared with respect to y . Let F : R k + l → R be globallysubanalytic. Then there are t ∈ N , a log-analytic function η : π ( C ) → R t anda globally subanalytic function I : R t × R r +1 → R such that F (cid:0) g ( x, y ) , . . . , g k ( x, y ) , log( h ( x, y )) , . . . , log( h l ( x, y )) (cid:1) = I ( η ( x ) , Y el r,C ( x, y )) for all ( x, y ) ∈ C . Let C be r -simple and let i : C → R be a function. Assume that there are t ∈ N , a log-analytic function η : π ( C ) → R t and a globally subanalytic function I : R t × R r +1 → R such that i ( x, y ) = I ( η ( x ) , Y el r,C ( y )) for all ( x, y ) ∈ C . Then there is a definable cell decomposition D of C suchthat i | D is purely r -log-analytically prepared with respect to y for every simple D ∈ D . Proof:
We do induction on r . Let z := ( z , . . . , z t ) and w := ( w , . . . , w r ). We set w ′ := ( w , . . . , w r ). r = 0: We prepare I globally subanalytically with respect to the coordinate w and find a globally subanalytic cell decomposition E of R t × R > such that forevery E ∈ E the restriction I | E is globally subanalytic prepared with respectto w . We find a definable cell decomposition A of C such that for every simple15 ∈ A there is E A ∈ E such that ( η ( x ) , y ) ∈ E A for every ( x, y ) ∈ A . Fix asimple A ∈ A . Since A is simple and y is plugged in for w we get that E A issimple with respect to w and therefore by Remark 2.10 in combination withCorollary 2.22 that I | E A is elementarily globally subanalytic prepared withrespect to w . Hence again by Remark 2.10 we have that I | E A = a ( z ) | w | q v (cid:0) b ( z ) | w | p , . . . , b s ( z ) | w | p s (cid:1) where a, b , . . . , b s are globally subanalytic functions on π ( E A ). We obtain i | C ( x, y ) = a ( η ( x )) | y | q v (cid:0) b ( η ( x )) | y | p , . . . , b s ( η ( x )) | y | p s (cid:1) and are done. r − → r : We prepare I globally subanalytically with respect to the coordinate w and find a globally subanalytic cell decomposition E of R t × R > × R r suchthat for every E ∈ E the restriction I | E is globally subanalytic prepared withrespect to w . We find a definable cell decomposition A of C such that forevery simple A ∈ A there is E A ∈ E such that ( η ( x ) , Y el r,C ( y )) ∈ E A for every( x, y ) ∈ A . Fix a simple A ∈ A . Since A is simple and y is plugged in for w we get that E A is simple with respect to w and therefore by Remark 2.10 incombination with Corollary 2.22 that I | E A is elementarily globally subanalyticprepared with respect to w . Hence again by Remark 2.10 we have that I | E A = a ( z, w ′ ) | w | q v (cid:0) b ( z, w ′ ) | w | p , . . . , b s ( z, w ′ ) | w | p s (cid:1) where a, b , . . . , b s are globally subanalytic functions. Denoting by c one ofthese functions we have to deal with c ( η ( x ) , y , . . . , y r ). Using composition ofpower series we are done with the following claim. Claim:
Let J : R t × R r → R be globally subanalytic. Then there is a definablecell decomposition B of A such that for every simple B ∈ B the function j : B → R , ( x, y ) J (cid:0) η ( x ) , y , . . . , y r (cid:1) is purely r -log-analytically prepared. Proof of the claim:
Since C is r -simple we get that A is r -simple. Set b A := (cid:8)(cid:0) x, − / log( y ) (cid:1) (cid:12)(cid:12) ( x, y ) ∈ A (cid:9) . Then b A is ( r − b J : R t × R r → R , ( z, w ′ ) α ( z, − /w , − w , w , . . . , w r ) , w < , if0 , else . Then for ( x, y ) ∈ A we have j ( x, y ) = b J (cid:16) η ( x ) , Y el r − , b A (cid:0) − / log( y ) (cid:1)(cid:17) . b j : b A → R , ( x, y ) b J (cid:0) η ( x ) , Y el r − , b A ( y ) (cid:1) we are done. (cid:4) Claim (cid:4)
Let f : R n × R > → R be log-analytic of order r . Then there is a definable celldecomposition C of X such that for every simple C ∈ C the cell C is r -simpleand f | C is purely r -log-analytically prepared with respect to y . Proof:
We do induction on the log-analytic order r of f . r = 0: Then f is piecewise globally subanalytic and we are done by Remark2.10. < r → r : It is enough to consider the following situation. Let g , . . . , g k : R n × R → R , h , . . . , h l : R n × R → R > be log-analytic functions of order atmost r − F : R k + l → R be globally subanalytic such that f = F ( g , . . . , g k , log( h ) , . . . , log( h l )) . Applying the inductive hypothesis and Corollary 2.28 in combination withCorollary 2.20 we find a definable cell decomposition C of R n × R > such thatevery simple C ∈ C is r -simple and for every such C there is t ∈ N , a log-analytic function η : π ( C ) → R t of order at most r and a globally subanalyticfunction E : R t × R r +1 → R , ( z, w ) E ( z, w ) such that f | C = E ( η ( x ) , Y el r,C ( x, y ))for all ( x, y ) ∈ C . By Proposition 2.29 we can refine the cell decompositionsuch that the assertion follows. (cid:4) Note that the above result extends the Expansion Theorem of [4, Section 4]to log-analytic functions.
Let f : R × R > → R , ( x, y ) f ( x, y ) , be log-analytic. Assume that for every x ∈ R n we have lim y ց f ( x, y ) ∈ R . Then the function h : R n → R , x lim y ց f ( x, y ) , is log-analytic. roof: By Theorem 2.30 we find a definable cell decomposition C of R n × R > such thatfor every simple C ∈ C the cell C is r -simple and f | C is purely r -log-analyticallyprepared with respect to y . Let C ∈ C be a simple cell. Set g := f | C and let J = (cid:0) r, Y el r,C , a, q, s, v, b, P (cid:1) be a purely r -preparing tuple for g . Then g ( x, y ) = a ( x ) |Y el r,C ( y ) | ⊗ q v (cid:16) b ( x ) |Y el r,C ( y ) | ⊗ p , . . . , b s ( x ) |Y el r,C ( y ) | ⊗ p s (cid:17) for ( x, y ) ∈ C . By the assumption, Remark 2.23 and Definition 2.1 we see that A : π ( C ) → R , x lim y ց a ( x ) |Y el r,C ( y ) | ⊗ q , and, for j ∈ { , . . . , s } , B i : π ( C ) → [ − , , x lim y ց b j ( x ) |Y el r,C ( y ) | ⊗ p i , are well defined and log-analytic. We obtain that for x ∈ π ( C ) h ( x ) = A ( x ) v (cid:16) B ( x ) , . . . , B s ( x ) (cid:17) . Hence h is log-analytic on π ( C ). By Remark 2.16 we get that h is log-analytic. (cid:4) With the above theorem we are able to establish Theorem A.
Theorem A
Let U ⊂ R n be definable and open and let f : U → R be log-analytic. Let i ∈ { , . . . , n } be such that f is differentiable with respect to the variable x i on U . Then ∂f /∂x i is log-analytic. Proof:
We may assume that f is differentiable with respect to the last variable x n .We have to show that ∂f /∂x n is log-analytic. Let e n := (0 , . . . , , ∈ R n bethe n th unit vector. We define V := (cid:8) ( x, y ) ∈ U × R > (cid:12)(cid:12) x + y e n ∈ U } and F : V → R , ( x, y ) f ( x + y e n ) − f ( x ) y . Then F is log-analytic. Since ∂f∂x n ( x ) = lim y ց F ( x, y )for x ∈ U we are done by Theorem 3.1. (cid:4) .2 Strong quasianalyticity - Proof of Theorem B Let U ⊂ R n be open and let g : U → R be a function. Let a ∈ U . The function g is N -flat at a if g is C N at a and all partial derivatives of g of order atmost N vanish in a . The function g is flat at a if g is C ∞ at a and all partialderivatives of g vanish in a .The asymptotic behaviour of log-analytic function on simple sets implied bythe elementary preparation gives the following (see [9] for the correspondingresult in polynomially bounded o-minimal structures). Let f : R n × R → R , ( x, y ) f ( x, y ) , be a log-analytic function. Then there is N ∈ N such that the following holds for every x ∈ R n : If f ( x, − ) is N -flat at y = 0 then f ( x, − ) vanishes identically on some open interval around ∈ R . Proof:
By also considering the function f ( x, − y ) it is enough to show that the follo-wing holds. There is N ∈ N such that for every x ∈ R n with f ( x, − ) being N -flat at y = 0 we have f ( x, y ) = 0 for all y ∈ ]0 , ε x [ for some ε x ∈ R > .Let r be the log-analytic order of f . By Theorem 2.30 we find a definable celldecomposition C of R n × R such that for every simple C ∈ C the cell C is r -simple and f | C is purely r -log-analytically prepared with respect to y . Let C ∈ C be simple and let J = (cid:0) r, Y el r,C , a, q, s, v, b, P (cid:1) be a purely r -preparing tuple for f | C . We show that a vanishes identically.Choose N C ∈ N such that N C ≥ q . Let x ∈ π ( C ). If f ( x, − ) is N C -flat at y = 0 then f ( x, − ) = o ( y N C ) at y = 0. But |Y el r,C | ⊗ q /y N C = o (1) by Remark2.22. Hence necessarily a ( x ) = 0.By Remark 2.16 we are done by taking N := max { N C | C ∈ C simple } . (cid:4) With Proposition 3.2 we can prove Theorem B using familiar connectivityarguments.
Theorem B
Let U ⊂ R n be a definable domain and let f : U → R be a log-analytic function.Then there is N ∈ N with the following property. If f is C N and if there is a ∈ U such that f is N -flat at a then f vanishes identically. Proof:
Let V := (cid:8) ( x, z, t, y ) ∈ U × S n − × R × R (cid:12)(cid:12) x + ( y − t ) z ∈ U (cid:9) F : V × R , ( x, z, t, y ) f (cid:0) x + ( y − t ) z (cid:1) . Then F is log-analytic. By Proposition 3.2 there is N ∈ N such that thefollowing holds for every ( x, z, t ) ∈ U × S n − × R such that ( x, z, t, ∈ V .If F ( x, z, t, − ) is N -flat at y = 0 then F ( x, z, t, − ) vanishes identically on anopen interval around 0 ∈ R .We assume that f is C N . Note that then F is C N . Let a ∈ U such that f is N -flat at a . We show that this implies that f vanishes identically and aredone. We start with the following. Claim:
There is r ∈ R > such that f vanishes identically on B ( a, r ). Proof of the claim:
Let r ∈ R > be such that B ( a, r ) ⊂ U . Then W := B ( a, r ) × S n − × ] − r, r [ × ] − r, r [ ⊂ V. Given z ∈ S n − we have that F ( a, z, , − ) is N -flat at y = 0. Then by above F ( a, z, , − ) vanishes on some open interval around 0. Fix z ∈ S n − and let A z be the set of all t ∈ ] − r, r [ such that F ( a, z, t, − ) vanishes identically onsome open interval around 0. Then A z = ∅ since 0 ∈ A z by above. Clearly A z is open. Let t ∈ A z ∩ ] − r, r [. Then F ( a, z, t, − ) is N -flat at 0. Hence by above F ( a, z, t, − ) vanishes identically on some open interval around 0. Therefore A z is closed in ] − r, r [. Since intervals are connected we obtain that A z = ] − r, r [and hence that F ( a, z, − , − ) vanishes identically on ] − r, r [ × ] − r, r [. Since z ∈ S n − is arbitrary we get that f vanishes identically on B ( a, r ). (cid:4) Claim
Let X be the set of all x ∈ U such that f vanishes identically on some openball around x . Then X = ∅ since a ∈ X by the above. Clearly X is open. Let x ∈ X ∩ U . Then f is N -flat at x . Again by the claim we get that x ∈ X .Therefore X is closed in U . Since U is connected we obtain that X = U andhence that f vanishes identically on U . (cid:4) With our result on pure preparation of log-analytic functions on simple setswe can establish the parametric version of Tamm’s theorem for log-analyticfunctions. For this we adapt the reasoning of Van den Dries and Miller [4].The theorem below deals with the case of a parameterized family of unaryfunctions where the adaptions are most extensive.
Let f : R n × R → R , ( x, y ) f ( x, y ) , be log-analytic. Then there is N ∈ N such that the following holds for all x ∈ R n : If f ( x, − ) is C N at then f ( x, − ) is real analytic at . roof: Let r ∈ N be the log-analytic order of f . By Theorem 2.30 we find a definablecell decomposition C of R n × R such that for every simple C ∈ C the cell C is r -simple and f | C is purely r -log-analytically prepared with respect to y . Fix a simple cell C ∈ C and set η x := sup C x for x ∈ π ( C ). Let J = (cid:0) r, Y el r,C , a, q, s, v, b, P (cid:1) be a purely r -preparing tuple for g := f | C . We set Y := Y el r,C . Let P α ∈ N s c α X α be the power series expansion of v . LetΓ := (cid:8) α ∈ N s (cid:12)(cid:12) t P α + q ∈ N × { } r (cid:9) and Γ := N s \ Γ . Set v := P α ∈ Γ c α X α and v := P α ∈ Γ c α X α . For l ∈ { , } let g l : C → R , ( x, y ) a ( x ) |Y ( y ) | ⊗ q v l (cid:0) b ( x ) |Y ( y ) | ⊗ p , . . . , b s ( x ) |Y ( y ) | ⊗ p s (cid:1) . Then g , g are log-analytic and g = g + g . For k ∈ N letΓ ,k := (cid:8) α ∈ N s (cid:12)(cid:12) t P α + q = ( k, , . . . , (cid:9) ⊂ Γ and d k : π ( C ) → R , x a ( x ) X α ∈ Γ k c α s Y i =1 b i ( x ) α i . Then g ( x, y ) = ∞ X k =0 d k ( x ) y k for ( x, y ) ∈ C . The series to the right converges absolutely on C and therefore g extends to a well defined extensionˆ g : ˆ C := (cid:8) ( x, y ) ∈ R n +1 (cid:12)(cid:12) x ∈ π ( C ) , − η x < y < η x (cid:9) → R , ( x, y ) ∞ X k =0 d k ( x ) y k , such that ˆ g ( x, − ) is real analytic at 0 for every x ∈ π ( C ). Claim 1:
The function ˆ g is log-analytic. Proof of Claim 1:
Clearly ˆ C is definable. We show that ˆ g is log-analyticon ˆ C ∩ ( R n × R < ) and on ˆ C ∩ ( R n × { } ) and are done. For the formerlet Γ ,e := S k even Γ ,k and Γ ,o := S k odd Γ ,k . Set v ,e := P α ∈ Γ ,e c α X α and v ,e := P α ∈ Γ ,o c α X α . Then for ( x, y ) ∈ ˆ C with y < g ( x, y ) = a ( x ) |Y ( − y ) | ⊗ q (cid:16) v ,e (cid:0) b ( x ) |Y ( − y ) | ⊗ p , . . . , b s ( x ) |Y ( − y ) | ⊗ p s (cid:1) − v ,o (cid:0) b ( x ) |Y ( − y ) | ⊗ p , . . . , b s ( x ) |Y ( − y ) | ⊗ p s (cid:1)(cid:17) g ( x,
0) = lim y ց g ( x, y )for x ∈ π ( C ). We are done by Theorem 3.1. (cid:4) Claim 1
Set ˆ g : ˆ C → R , ( x, y ) f ( x, y ) − ˆ g ( x, y ) . Then ˆ g is log-analytic by Claim 1 and ˆ g | C = g . Fix x ∗ ∈ π ( C ). LetΛ := (cid:8) t P α + q (cid:12)(cid:12) α ∈ Γ (cid:9) . Then Λ ⊂ Q r +1 \ ( N × { } r ). For λ ∈ Λ, letΓ ,λ := (cid:8) α ∈ N s (cid:12)(cid:12) t P α + q = λ (cid:9) and e x ∗ ,λ := a ( x ∗ ) X α ∈ Γ ,λ c α s Y i =0 b i ( x ∗ ) α i ∈ R . Then ˆ g ( x ∗ , − ) = X λ ∈ Λ e x ∗ ,λ |Y | ⊗ λ on ]0 , η x ∗ [. Let Λ x ∗ := { λ ∈ Λ | e x ∗ ,λ = 0 (cid:9) . If Λ x ∗ = ∅ then ˆ g ( x ∗ , − ) = 0 on ]0 , η x ∗ [. If Λ x ∗ = ∅ there is µ x ∗ = ( µ x ∗ , , . . . , µ x ∗ ,r ) ∈ Λ x ∗ such that |Y | ⊗ λ = o ( |Y | ⊗ µ x ∗ ) for all λ ∈ Λ x ∗ with λ = µ x ∗ . Claim 2:
Assume that Λ x ∗ = ∅ . Let M ∈ N be such that f ( x ∗ , − ) is C M at 0.Then µ x ∗ , ≥ M . Proof of Claim 2:
Assume that µ x ∗ , < M . Case 1: µ x ∗ , ∈ N .Then m := µ x ∗ , + 1 ≤ M . Differentiating g m -times with respect to y we seewith Proposition 2.24 (note that µ x ∗ = 0) that there is β = ( − , β , . . . , β r ) ∈ Q r +1 such that lim y ց ∂ m g /∂y m ( x ∗ , y ) |Y ( y ) | ⊗ β ∈ R ∗ . Since ˆ g ( x ∗ , − ) = f ( x ∗ , − ) − ˆ g ( x ∗ , − ) is C M at 0 we obtain thatlim y ց ∂ m ˆ g ∂y m ( x ∗ , y ) = ∂ m ˆ g ∂y m ( x ∗ , ∈ R , which contradicts that ˆ g ( x ∗ , − ) extends g ( x ∗ , − ).22 ase 2: µ x ∗ , / ∈ N .Then m := ⌈ µ x ∗ , ⌉ ≤ M . Differentiating g m -times with respect to y we seewith Proposition 2.24 (note that µ x ∗ = 0) that there is β = ( β , β , . . . , β r ) ∈ Q r +1 with β < y ց ∂ m g /∂y m ( x ∗ , y ) |Y | ⊗ β ∈ R ∗ . But ˆ g ( x ∗ , − ) = f ( x ∗ , − ) − ˆ g ( x ∗ , − ) is C M at 0. We get the same contradictionas in Case 1. (cid:4) Claim 2
Claim 3:
Let M ∈ N be such that f ( x ∗ , − ) is C M at 0. Then ˆ g ( x ∗ , − ) is( M − Proof of Claim 3:Case 1: Λ x ∗ = ∅ .Then ˆ g ( x ∗ , − ) = 0 on ]0 , η x ∗ [ and we are clearly done. Case 2: Λ x ∗ = ∅ .By Claim 1 we obtain that µ x ∗ , ≥ M . Hence we obtain by Proposition 2.24and Remark 2.23 that lim y ց ∂ m g ∂y m ( x ∗ , y ) = 0for all m ∈ { , . . . , M − } . Since ˆ g ( x ∗ , − ) = f ( x ∗ , − ) − ˆ g ( x ∗ , − ) is C M at 0and since ˆ g ( x ∗ , − ) extends g ( x ∗ , − ) we are done. (cid:4) Claim 3
Since ˆ g is log-analytic we find by Proposition 3.2 some K C ∈ N such that thefollowing holds for every x ∈ π ( C ). If ˆ g ( x, − ) is K C -flat at y = 0 then ˆ g ( x, − )vanishes identically on some open interval around 0. Set N C := K C +1. Assumethat f ( x, − ) is C N C at 0. Then by Claim 3 ˆ g ( x, − ) is K C -flat and hence by theabove that f ( x, − ) = ˆ g ( x, − ) on some open interval around 0. Since ˆ g ( x, − )is real analytic at 0 we get that f ( x, − ) is real analytic at 0. By Remark 2.16we are done by taking N := max { N C | C ∈ C simple } . (cid:4) Let f : R n × R → R , ( x, y ) f ( x, y ) , be log-analytic such that f ( x, − ) is realanalytic at for every x ∈ R n . Then there is a definable cell decomposition B of R n such that B → R , x d k /dy k f ( x, , is real analytic for every B ∈ B and every k ∈ N . Proof:
Using the notation of the previous proof we have f ( x, y ) = ˆ g ( x, y ) for all( x, y ) ∈ C where C is a simple cell of the constructed cell decomposition C .23ince functions definable in R an , exp are piecewise real analytic (see [5, Section4]) we find a cell decomposition D of π ( C ) ⊂ R n such that the coefficient a and the base functions b , . . . , b s are real analytic on every D ∈ D . Hence oneach D ∈ D the coefficients d k of ˆ g are real analytic for every k ∈ N . Since d k /dy k f ( x,
0) = k ! d k ( x ) we are done by Remark 2.16. (cid:4) Now we can finish as in [4, Section 5]. For the reader’s convenience we give thedetails.Let U ⊂ R n be open and let g : U → R be a function. Let a ∈ U . Let k ∈ N .Then g is called k -times Gateaux-differentiable or G k at a if y g ( a + yz )is C k at y = 0 for every z ∈ R n and z (cid:0) d k g ( a + yz ) /dy k (cid:1) (0) is given by ahomogeneous polynomial in z of degree k . The function g is called G ∞ at a if g is G k at a for every k ∈ N . The following holds: [4, Proposition 2.2]) Let U ⊂ R n be open and let g : U → R be a function. Let a ∈ U . The followingare equivalent.(i) The function g is real analytic at a .(ii) The function g is G ∞ at a and there is ε ∈ R > such that for every z ∈ R n with | z | ≤ the function y g ( a + yz ) is defined and realanalytic on ] − ε, ε [ . Let f : R n × R m → R , ( x, u ) f ( x, u ) , be a log-analytic function and let k ∈ N .Then there is a log-analytic function w k : R n × R m × R m → R , ( x, u, v ) w k ( x, u, v ) , such that the following is equivalent for every ( x, u ) ∈ R n × R m .(i) The function f ( x, − ) is G k at u .(ii) It is w k ( x, u, v ) = 0 for every v ∈ R m . Proof:
For k ∈ N let W k be the set of all ( x, u ) ∈ R n × R m such that y f ( x, u + yv )is k -times differentiable in 0 for every v ∈ R m . We defineΦ k : R n × R m × R m → R , ( x, u, v ) d k f ( x,u + yv ) dy k (0) , ( x, u ) ∈ W k , if1 , ( x, u ) / ∈ W k . Then Φ k is log-analytic by Theorem A. Let ν ( k ) be the dimension of the realvector space of homogeneous real polynomials of degree k in the variables V :=24 V , . . . , V m ) and let M ( V ) , . . . , M ν ( k ) ( V ) be the homogeneous monomials ofdegree k in V . There are points p k, , . . . , p k,ν ( k ) ∈ R m and linear functions a , . . . , a ν ( k ) : R ν ( k ) → R such that for all s := ( s , . . . , s ν ( k ) ) ∈ R ν ( k ) P k ( s, V ) := ν ( k ) X j =1 a j ( s ) M j ( V ) ∈ R [ V ]is the unique homogeneous polynomial of degree k with P k ( s, p k,i ) = s i for all i ∈ { , . . . , ν ( k ) } (see [4, Section 2.3]). Setˆ w k : R n × R m × R m → R , ( x, u, v ) P k (cid:0) Φ k ( x, u, p k, ) , . . . , Φ k ( x, u, p k,ν ( k ) ) , v (cid:1) and w k := ˆ w k − Φ k . Then w k is log-analytic. We show that it fulfils the requi-rements.( i ) ⇒ ( ii ): Let ( x, u ) ∈ R n × R m be such that f ( x, − ) is G k at u . Then ( x, u ) ∈ W k and v Φ k ( x, u, v ) is a homogeneous polynomial of degree k . By thedefinition of ˆ w k we have ˆ w k ( x, u, p k,i ) = Φ k ( x, u, p k,i ) for all i ∈ { , . . . , ν ( k ) } .By the uniqueness of P k we obtain thatˆ w k ( x, u, v ) = P k (cid:0) Φ k ( x, u, p k, ) , . . . , Φ k ( x, u, p k,ν ( k ) ) , v (cid:1) = Φ k ( x, u, v )and therefore w k ( x, u, v ) = 0 for all v ∈ R m .( ii ) ⇒ ( i ): Let ( x, u ) ∈ R n × R m be such that w k ( x, u, v ) = 0 for all v ∈ R m .Then Φ k ( x, u, v ) = ˆ w k ( x, u, v ) for all v ∈ R m and therefore v Φ k ( x, u, v ) isa homogeneous polynomial of degree k . Since k ≥ x, u ) ∈ W k and consequently f is G k at u . (cid:4) Let f : R n × R m → R , ( x, u ) f ( x, u ) , be a log-analytic function. Then thereis N ∈ N such that the following holds for every ( x, u ) ∈ R n × R m . If f ( x, − ) is G N at u then f ( x, − ) is G ∞ at u . Proof:
By Theorem 3.3 there is K ∈ N such that the following holds for every( x, u, v ) ∈ R n × R m × R m . If y f ( x, u + yv ) is C K at 0 then y f ( x, u + yv )is real analytic at 0. Letting as in the previous proof W k for k ∈ N to be theset of all ( x, u ) ∈ R n × R m such that y f ( x, u + yv ) is k -times differen-tiable at 0 for every v ∈ R m we get that W k = W K for all k ≥ K . Lettingalso Φ k for k ∈ N be as in the previous proof and dealing with Φ , . . . , Φ K − we find by Corollary 3.4 a definable cell decomposition C of R n × R m × R m such that Φ k | C is real analytic for every C ∈ C and all k ∈ N . Constructing w k : R n × R m × R m → R , ( x, u, v ) → w k ( x, u, v ) for k ∈ N as in the previous25roof we see that w k | C is real analytic for every k ∈ N . By Tougeron [11] (seealso [4, Proposition 1.6]) we find for every C ∈ C some N C ∈ N such that \ k ∈ N (cid:8) ( x, u, v ) ∈ C (cid:12)(cid:12) w k ( x, u, v ) = 0 (cid:9) = \ k ≤ N C (cid:8) ( x, u, v ) ∈ C (cid:12)(cid:12) w k ( x, u, v ) = 0 (cid:9) . Let N := max { N C | C ∈ C} . Then \ k ∈ N (cid:8) ( x, u, v ) ∈ R n × R m × R m (cid:12)(cid:12) w k ( x, u, v ) = 0 (cid:9) == \ k ≤ N (cid:8) ( x, u, v ) ∈ R n × R m × R m (cid:12)(cid:12) w k ( x, u, v ) = 0 (cid:9) . Hence for every ( x, u ) ∈ R n × R m we have that f ( x, − ) is G ∞ at u if and onlyif w k ( x, u, v ) = 0 for all k ∈ N and all v ∈ R m if and only if w k ( x, u, v ) = 0 forall k ∈ { , . . . , N } and all v ∈ R m if and only if f ( x, − ) is G N at u . (cid:4) Theorem C
Let f : R n × R m → R , ( x, y ) f ( x, y ) , be a log-analytic function. Then thereis N ∈ N such that the following holds for every ( x, y ) ∈ R n × R m . If f ( x, − ) is C N at y then f ( x, − ) is real analytic at y . Proof:
By Proposition 3.7 there is N ∈ N such that if f ( x, − ) is G N at y then f ( x, − ) is G ∞ at y . Let F : R n × R m × R m × R → R , ( x, y, z, t ) f ( x, y + tz ) . By Theorem 3.4 there is N ∈ N such that if F ( x, y, z, − ) is C N at 0 then F ( x, y, z, − ) is real analytic at 0. Taking N := max { N , N } we are done byFact 3.5. (cid:4) Let f : R n × R m → R , ( x, y ) f ( x, y ) , be a log-analytic function. Then theset of all ( x, y ) ∈ R n × R m such that f ( x, − ) is real analytic at y is definable. References (1) R. Cluckers and D. Miller: Stability under integration of sums of products of realglobally subanalytic functions and their logarithms.
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