A remark on the Lion–Rolin Preparation Theorem for LA-functions

ANNALES

POLONICI MATHEMATICI LXXI.2 (1999)

A remark on the Lion–Rolin Preparation Theorem for LA-functions

by Wies law Paw lucki and Artur Pie ¸kosz (Krak´ ow)

Abstract. A correct formulation of the Lion–Rolin Preparation Theorem for loga- rithmic-subanalytic functions (LA-functions) is given.

In [2] Lion and Rolin give an explicit description of functions on R

(n ∈ Z, n > 0), called by them LE-functions, defined as finite composi- tions of globally subanalytic functions with logarithmic and with exponential functions. This enables them to obtain the fundamental results of van den Dries, Macintyre and Marker [1] without making use of model theory. One important step in their study is their Preparation Theorem for LA-functions.

To quote this theorem we first recall some basic definitions from [2].

If F is any family of real functions on R

, a subset E of R

is called an F -set if

E = [

i∈I

\

j∈J

E

where I and J are finite and for each (i, j) ∈ I × J , E

= {φ

> 0} or E

= {φ

= 0} or E

= {φ

< 0}, with some φ

∈ F .

A subset C of R

= {(x, y) | x ∈ R

, y ∈ R} is called an F-cylinder if either

C = {(x, y) | x ∈ B, y = φ(x)}

where B is an F -set in R

and φ ∈ F , or

C = {(x, y) | x ∈ B, φ(x) < y < ψ(x)},

where B is an F -set in R

, φ, ψ ∈ F ∪ {−∞}

∪ {∞}

and φ(x) < ψ(x) for each x ∈ B.

Let f : R

= R

× R → R. We say that

1991 Mathematics Subject Classification: Primary 32B20; Secondary 33B10.

Key words and phrases: subanalytic function, logarithmic-subanalytic function, redu- cible.

[195]

196 W. P a w l u c k i and A. P i ¸ e k o s z

(1) f is an LA-function of type 0 if f can be represented as f (x, y) = F (a

(x), . . . , a

(x), y),

where m ∈ Z, m > 0, F is a globally subanalytic function on R

, and a

are LE-functions on R

;

(2) f is an LA-function of type r, r ≥ 1, if f can be represented as f (x, y) = F (f

(x, y), . . . , f

(x, y), log |f

(x, y)|, . . . , log |f

(x, y)|), where F is globally subanalytic, m, l ∈ Z, m, l > 0, and f

are LA-functions of type r − 1.

An LA-function f : R

→ R of type r is called reducible if there exists a finite covering C of R

by LE-cylinders such that, on each C ∈ C, f can be represented in the form

f (x, y) = |y

|

. . . |y

|

A(x)U (x, y

, . . . , y

),

where y

= y − θ

(x), y

= log |y

| − θ

(x), . . . , y

= log |y

| − θ

(x), with some LE-functions θ

such that |y

| ≤ M |θ

(x)| on C, A is an LE-function, α

∈ Q (j = 0, . . . , r), and

U (x, y

, . . . , y

) = V (ψ(x, y)), where

ψ(x, y)

=



φ

(x), . . . , ψ

(x), |y

|

a

(x) , . . . , |y

|

a

(x) , . . . , b

(x)

|y

|

, . . . , b

(x)

|y

|

 , with some LE-functions φ

, a

, b

, positive integers p

, and an analytic non- vanishing function V of constant sign in a neighbourhood of the compact set ψ(C) in (P

)

, where P

denotes the real projective line.

Lion and Rolin formulate the following Preparation Theorem for LA- functions [2, Th´ eor` eme 2]: Every LA-function is reducible.

Our goal here is to observe that this formulation requires some correction.

To see this, consider the function f : R → R defined as follows:

f (y) =  y

= e

if y ∈ (0, 1), 0 if y 6∈ (0, 1).

Of course, f is an LA-function of type 1. However, it is not reducible in the above sense. If it were, we would have the following equality in some interval (0, ε) (ε > 0):

f (y) = |y

|

|y

|

V (|y

|

, |y

|

, |y

|

, |y

|

),

where y

= y − θ

, y

= log |y

| − θ

, p

, p

are positive integers, α

, α

∈ Q and V is an analytic positive function in a neighbourhood of ψ(0, ε) in (P

)

, where

ψ(y) = (|y

|

, |y

|

, |y

|

, |y

|

).

The Lion–Rolin Preparation Theorem 197

If θ

6= 0, then f would be subanalytic near 0, which is not the case, so suppose that θ

= 0. Then

f (y) = |y|

|y

|

V (|y| e

, |y

|

),

where e V is analytic in a neighbourhood of (0, 0) and e V (0, 0) > 0.

If α

6= 0 or α

6= 0, the right-hand side would tend to 0 or ∞ as y tends to 0; a contradiction. Hence,

f (y) = e V (|y|

, |y

|

) and y log y = (log e V )(|y|

, |y

|

) for y ∈ (0, ε

). Expanding log e V with respect to the first variable, we have

y log y =

∞

X

ν=k

C

(|y

|

)|y|

= C

(|y

|

)|y|

×



1 + |y|

C

(|y

|

)

∞

X

ν>k

C

(|y

|

)|y|

 . Hence y

(log y)[C

(|y

|

)]

tends to 1 as y tends to 0; this is asymptotically equivalent to

y

(log y)|y

|

for some l > 0, so tends to 0 or ∞.

This example indicates that in order to obtain a correct formulation of the theorem one should allow ψ in the definition of reducibility in a more general form, viz.

ψ(x, y) = (a

(x)|y

|

. . . |y

|

, . . . , a

(x)|y

|

. . . |y

|

), where β

∈ Q.

The proof after this modification is that of [2].

References

[1] L. v a n d e n D r i e s, A. M a c i n t y r e and D. M a r k e r, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. 140 (1994), 183–205.

[2] J.-M. L i o n et J.-P. R o l i n, Th´ eor` eme de pr´ eparation pour les fonctions logarith- mico-exponentielles, Ann. Inst. Fourier (Grenoble) 47 (1997), 859–884.

Wies law Paw lucki Institute of Mathematics Jagiellonian University Reymonta 4

30-059 Krak´ ow, Poland E-mail: pawlucki@im.uj.edu.pl

Artur Pi¸ ekosz Institute of Mathematics Cracow University of Technology Warszawska 24 31-155 Krak´ ow, Poland E-mail: pupiekos@cyf-kr.edu.pl

Re¸ cu par la R´ edaction le 24.9.1998