RECTILINEARIZATION AND UNIFORMIZATION OF k-SUBANALYTIC SETS AND FUNCTIONS

by Artur Pie‘kosz

IMUJ PREPRINT 1999/30 KRAKÓW

Abstract We prove rectilinearization and uniformization theorems for ksubanalytic (ℝk_{an}-definable) sets and functions using the Lion-Rolin formula. Parallel reasoning gives standard results for the subanalytic case.

1 Introduction.

The aim of this paper is to give a new proof of earlier known uniformization and rectilinearization theorems for subanalytic sets ([4], [1]) and to get generalizations of these theorems to a larger class of sets which we call k-subanalytic.

k-subanalytic sets are these which can be described (or defined) by compositions of globally subanalytic functions and power functions |x|^{λ} with λ ∈ k, where k is a given subfield of the field of real numbers ℝ.

AMS Classification: 32B20, 14P15, 32C05.

Key words: k-subanalytic, preparation theorem, rectilinearization, uniformization.

This class of sets was earlier studied by J.-Cl. Tougeron ([9]) and by L. van den Dries and C. Miller (for example [2, 6]).

In 1997 J.-M. Lion and J.-P. Rolin published a paper [5], where they gave an explicit formula for a globally subanalytic function using analytic functions and operations of division and taking roots. This theorem, called the preparation theorem for subanalytic functions is similar to the Weierstrass preparation theorem, and appeared in less explicit form in a paper of A. Parusiński [8].

We use the Lion-Rolin formula for k-subanalytic functions, which is an analog of the Parusiński-Lion-Rolin preparation theorem for subanalytic functions, and get uniformization and rectilinearization for k-subanalytic functions and sets. This gives a partial answer to a problem formulated by L. van den Dries and C. Miller in [3].

Parallel reasoning gives uniformization and rectilinearization theorems for the subanalytic case. In particular, we get a version of the characterization of subanalytic continuous functions obtained by E. Bierstone and P. Milman [1], and by A. Parusiński [7].

2 Notions.

2.1 General notions.

Let X be any set. Functions f, g : X → ℝ are called equivalent (notation: f ∼ g) on X, when there exists a constant M > 0 such that for x ∈ X M^{−1} · |f(x)| ≤ |g(x)| ≤ M · |f(x)|. A unit is a function equivalent to the constant function 1.

We assume that analytic manifolds are paracompact. If Z ⊂ M, where M is an analytic manifold, then reg Z is the set of these points of Z which are analytically regular (i.e. Z is an analytic manifold in a neighbourhood of any point of reg Z). By ℙ = ℙ(ℝ) we denote the 1-dimensional real projective space. A box is a cartesian product of (possibly degenerated) intervals.

2.2 Cylinders and cells.

Let X be any set and let F be a finite family of real functions on X. An F-set is any subset of X of the form

E = ⋃_{i=1}^{k} ⋂_{j=1}^{l} E_{ij}, (k, l ∈ ℕ)

where E_{ij} = {x ∈ X| g_{ij}(x) = 0} or E_{ij} = {x ∈ X| g_{ij}(x) > 0} or E_{ij} = {x ∈ X| g_{ij}(x) < 0} and g_{ij} ∈ F. Notice that F-sets form a boolean algebra.

A set C ⊂ X × ℝ is called an F-cylinder when it is of the form

(1) C = {(x, y) ∈ B × ℝ| g_{1}(x) < y < g_{2}(x)}, where B is an F-set in X and g_{1}, g_{2} ∈ F ∪ {−∞}^{X} ∪ {+∞}^{X}, with g_{1}(x) < g_{2}(x) for every x ∈ B,

or

(2) C = {(x, y) ∈ B × ℝ| y = g(x)}, where B is an F-set in X and g ∈ F. Then B is called the basis of C and the functions g_{1}, g_{2} (or the function

g) are called the generating functions of the cylinder C.

Lemma 2.2.1 Let F be an ℝ-algebra of functions on X. Then for every finite family C of F-cylinders in X × ℝ there exists a finite decomposition C′ of X × ℝ into F-cylinders compatible with C.

The proof is easy and left to the reader. □ Let, for every n ∈ ℕ, we have a family A_{n} of functions from ℝ^{n} into ℝ.

Denote A = ⋃_{n=0}^{∞} A.

An A-cell (more exactly: A_{n}-cell) in ℝ^{n+1} is such an A_{n}-cylinder K whose basis K′ is an A_{n−1}-cell in ℝ^{n} . ({0} is A_{−1}-cell in ℝ^{0}, where A_{−1} = ∅.)

Every A-cell K ⊂ ℝ^{n+1} is associated with a sequence K = K^{n+1}, K′ = K^{n}, ..., K^{0} where K^{j} is an A_{j−1}-cell in ℝ^{j} (j = n + 1, ..., 0) and

(1)K^{j+1} = {(x_{1}, ..., x_{j+1}) ∈ K^{j}×ℝ|t_{j+1,1}(x_{1}, ..., x_{j}) < x_{j+1} < t_{j+1,2}(x_{1}, ..., x_{j})},

where t_{j+1,1} < t_{j+1,2} on K^{j}, or

(2)K^{j+1} = {(x_{1}, ..., x_{j+1}) ∈ K^{j}×ℝ|t_{j+1,1}(x_{1}, ..., x_{j}) = x_{j+1} = t_{j+1,2}(x_{1}, ..., x_{j})},

where t_{j+1,1} = t_{j+1,2} on K^{j},

with t_{j+1,i} ∈ A_{j} ∪ {−∞}^{ℝ^{j}} ∪ {+∞}^{ℝ^{j}}(i = 1, 2), and t_{j+1,i} ≡ −∞ or t_{j+1,i} ≡ +∞ can hold only in the case (1). The sequence {t_{j,i}}_{j=1,...,n+1}^{i=1,2} is called the sequence of generating functions of the A-cell K.

We say that K is full-dimensional, when K′ is full-dimensional and K is of the form (1). A-cell {0} is full-dimensional in ℝ^{0}.

We say that K is an analytic cell, when each of the functions {t_{j,i}|K^{j−1}}_{j=1,...,n+1}^{i=1,2} is a restriction of some analytic function in a neighbourhood of K^{j−1} (or, equivalently, its generating functions are analytic on respective bases as submanifolds).

If the family A is fixed, we will speak about cells and cyliders instead of A-cells and A_{n}-cylinders.

2.3 Subanalytic notions.

A subset Z of an analytic manifold M is called semianalytic in M, if for every x ∈ M there exists an open neighbourhood U of a in M such that Z ∩ U is an A(U)-set in U, where A(U) is the algebra of all analytic functions on U.

A subset Z ⊂ ℝ^{n} is called globally subanalytic in ℝ^{n} when it is an image Z = π(A) of the natural projection π of some set A ⊂ ℝ^{n+k} semianalytic in ℙ^{n+k} (ℝ is embedded in ℙ in a standard way). A mapping h : ℝ^{n} ⊃ X → Y ⊂ ℝ^{m} is globally subanalytic if its graph is globally subanalytic in ℝ^{n} × ℝ^{m}.

Denote

SAN_{n} = {f : ℝ^{n} → ℝ| f is globally subanalytic}.

We will often assume that SAN_{n} ⊂ SAN_{n+1}, treating functions from SAN_{n} as independent on the last variable.

It follows from the Parusiński-Lion-Rolin preparation theorem that SAN_{n}- sets in ℝ^{n} are exactly globally subanalytic sets. A consequence of this is the Gabrielov theorem of the complement.

We will use the notion of a blowing-up in the most standard sense: any mappping of the form

R_{i,j} : ℝ^{n} ∋ (x_{1}, ..., x_{i}, ..., x_{j}, ..., x_{n}) ↦ (x_{1}, ..., x_{i}, ..., x_{j} · x_{i}^{(j)}, ..., x_{n}) ∈ ℝ^{n},

where n ≥ 2, 1 ≤ i, j ≤ n and i ≠ j.

A power function is any composition of mappings of the form

P : ℝ^{n} ∋ (x_{1}, ..., x_{i}, ..., x_{n}) ↦ (x_{1}, ..., x_{i}^{γ}, ..., x_{n}) ∈ ℝ^{n},

where γ ∈ ℕ^{∗} = ℕ \ {0}.

A (subanalytic) normal crossing is a function on an open, globally subanalytic subset of some ℝ^{n} being a product of a monomial and an analytic and globally subanalytic unit. (We will drop the word ”subanalytic”.) A local normal crossing is a normal crossing with some analytic and globally subanalytic isomorphism substituted.

Remarks.

1) A composition of combinations of normal crossings is a combination of normal crossings.

2) A bounded quotient of normal crossings on a bounded open box in ℝ^{n} is a normal crossing.

A subanalytic modification is any mapping Φ : U → ℝ^{n}, where U is open and globally subanalytic in ℝ^{n}, and Φ is a composition of blowingsup, power functions and analytic isomorphisms of open sets in ℝ^{n} which are globally subanalytic. If U = ℝ^{n}, then we say that this modification is global. Subanalytic modifications are analytic and globally subanalytic mappings.

A semialgebraic modification is any mapping Φ : U → ℝ^{n}, where U is open and semialgebraic in ℝ^{n}, and Φ is a composition of blowings-up, power functions and semialgebraic Nash (analytico-algebraic) isomorphisms of open sets in ℝ^{n}. If U = ℝ^{n}, then the modification is global. Semialgebraic modifications are semialgebraic Nash mappings.

Lemma 2.3.1 (on boxes) Every compact box (not necessarily full-dimen-sional) K ⊂ ℝ_{+}^{n} is a union of images of subspaces ℝ^{dim K} ⊂ ℝ^{n} by a finite family of semialgebraic modifications m_{μ} : ℝ^{n} → ℝ^{n} which are combinations of local normal crossings. Moreover, there exists a compact box L ⊂ ℝ^{dim K} such that ⋃_{μ} m_{μ}(L) = K.

Proof. As we can make cartesian products of m_{μ}-s, it suffices to consider the case n = 1. A singleton is the image of the trivial subspace of ℝ by a translation. A nondegenerated interval [a, b] is the union of [a, a+\frac{3}{4}(b−a)) and (a+ \frac{1}{4}(b−a), b] which are images of ℝ by compositions of Nash isomorphisms, power functions, and linear isometries. (They are normal crossings in the variable y = \frac{x}{\sqrt{1+x^{2}}} .) The existence of a proper L is obvious. □

Let M be an analytic manifold of dimension n and let x ∈ M, E ⊂ M. A germ E_{x} is called subanalytic when there exists a chart ϕ in a neigh-

bourhood of x such that ϕ(E)_{ϕ(x)} is a germ of a globally subanalytic set in ℝ^{n}. (It does not depend on the choice of a chart.)

A set E is called subanalytic if for every x ∈ M its germ E_{x} is subanalytic. A mapping of analytic manifolds f : M → N is a subanalytic mapping if its graph is a subanalytic subset of M ×N. (A set E ⊂ ℝ^{n} is globally subanalytic if and only if it is subanalytic in ℙ^{n}.)

A function f : E → ℝ has a subanalytic graph in M × ℙ if and only if for every x ∈ M there exists a chart ϕ : U → ϕ(U) ⊂ ℝ^{n} in a neighbourhood of x such that ϕ×id_{ℝ}(graph f) is globally subanalytic in ℝ^{n} ×ℝ. Such functions will be called globally subanalytic on M.

A subanalytic modification into M is a subanalytic mapping π : ℝ^{n} ⊃ V → M such that \overline{im π} is contained in a domain of a chart ϕ and ϕ ∘ π is a subanalytic modification.

2.4 k-subanalytic notions.

For every real number λ, we define a function

| · |^{λ} : ℝ ∋ x ↦ \{ \begin{matrix}1, & x = 0, λ = 0 \\ 0, & x = 0, λ ≠ 0 \\ |x|^{λ}, & x ≠ 0. \end{matrix}

Notice that for λ ≥ 0 the function | · |^{λ} is continuous, and if λ > p ∈ ℕ, then | · |^{λ} is of class C^{p}. If x ∈ ℝ^{n} and λ ∈ ℝ^{n}, then |x|^{λ} denotes |x_{1}|^{λ_{1}} · ... · |x_{n}|^{λ_{n}}.

Let k be a fixed subfield of the real field ℝ. We put

PSk_{n} = {f : ℝ^{n} → ℝ| f is a composition of functions from SAN

and functions of the form |x|^{λ}(λ ∈ k)}.

We will often assume that PSk_{n} ⊂ PSk_{n+1} treating functions from PSk_{n} as independent on the last variable.

k-subanalytic sets in ℝ^{n} are PSk_{n}-sets. k-subanalytic functions with domains in ℝ^{n} are functions having k-subanalytic graphs. (ℚ-subanalytic sets and functions in ℝ^{n} are exactly globally subanalytic sets and functions.)

By the k-subanalytic version of the Lion-Rolin preparation theorem,

PSk_{n} = {f : ℝ^{n} → ℝ| f is k-subanalytic}.

A k-analytic function is a function V : U → ℝ whose domain is a ksubanalytic open set in ℝ^{n}, and being of the form V (x) = Ṽ (Θ(x)), where Θ(x) = (x_{1}, ..., x_{n}, |x_{σ(1)}|^{λ_{1}}, ..., |x_{σ(m)}|^{λ_{m}}), m ∈ ℕ, σ : {1, ..., m} → {1, ..., n} is any sequence, λ_{i} ∈ k_{+}^{∗} = {κ ∈ k| κ > 0}, for i = 1, ..., m and Ṽ is an analytic and globally subanalytic function in some neighbourhood of Θ(U) ⊂ ℝ^{n+m} .

Remark. k-analytic functions are continuous, analytic off {x ∈ ℝ^{n}| x_{1} · ... · x_{n} = 0}, and if all λ_{i} > p ∈ ℕ^{∗} then they are of class C^{p}.

Example. The function z(x, y) = ||x|^{π} + y|^{π} is a composition of ℝanalytic functions which is not ℝ-analytic.

A k-analytic mapping is a mapping Ω : ℝ^{n} ⊃ A → B ⊂ ℝ^{m}(m, n ∈ ℕ) whose combinational components are k-analytic functions. k-analytic mappings are continuous.

A k-analytic isomorphism is a bijective mapping which is k-analytic together with its inverse. k-analytic isomorphisms are k-subanalytic homeomorphisms of open sets.

A k-power function is any composition of mappings of the form

P_{i} : ℝ^{n} ∋ (x_{1}, ..., x_{n}) ↦ (x_{1}, ..., |x_{i}|^{γ}, ..., x_{n}) ∈ ℝ^{n},

where γ ∈ k_{+}^{∗}. k-power function is a k-analytic mapping.

A k-normal crossing is a function defined on an open, k-subanalytic subset of some ℝ^{n} and being a product of a monomial with exponents from k_{+} = {κ ∈ k| κ ≥ 0} and a k-analytic unit. A local k-normal crossing is a k-normal crossing with some analytic and globally subanalytic isomorphism substituted.

Remarks.

1) A composition of combinations of k-normal crossings is a combination of k-normal crossings. A combination of k-normal crossings and zero functions is a k-analytic mapping. Substitution of a combination of k-normal crossings and zero functions in a k-analytic mapping is a k-analytic mapping.

2) A bounded quotient of k-normal crossings on a bounded open box is a k-normal crossing.

3) The mapping ℝ^{n+1} ∋ (x, y) ↦ (|x|, |x|^{α} · y) ∈ ℝ^{n+1} (y ∈ ℝ, x ∈ ℝ^{n}, α ∈ k_{+}^{n}) is a composition of blowings-up and k-power functions.

4) The mapping ℝ^{1+1} ∋ (x, y) ↦ (|x·y|^{α}, y) ∈ ℝ^{1+1} (y ∈ ℝ, x ∈ ℝ, α ∈ k_{+}^{∗}) is a composition of a blowing-up and a k-power function.

A k-subanalytic modification is a mapping Φ : U → ℝ^{n}, where U is open and k-subanalytic, and Φ is a composition of blowings-up, k-power functions and k-analytic isomorphisms. If U = ℝ^{n}, then the modification is global. k-subanalytic modifications are k-subanalytic continuous mappings.

Let M be an analytic manifold of dimension n and let x ∈ M, E ⊂ M.

A germ E_{x} is called k-subanalytic if there exists a chart ϕ in a neighbourhood of x such that ϕ(E)_{ϕ(x)} is a germ of a k-subanalytic set in ℝ^{n}. (It does not depend on the choice of a chart.)

A set E is called locally k-subanalytic if for every x ∈ M its germ E_{x} is k-subanalytic. A mapping of analytic manifolds f : M → N is called locally k-subanalytic if its graph is locally k-subanalytic in M × N.

A function f : E → ℝ has a locally k-subanalytic graph in M × ℙ if and only if for every x ∈ M there exists a chart ϕ : U → ϕ(U) ⊂ ℝ^{n} in a neighbourhood of x such that ϕ × id_{ℝ}(graph f) is k-subanalytic in ℝ^{n} × ℝ. Such functions will be called k-subanalytic on M.

A k-subanalytic modification into M is a locally k-subanalytic mapping π : ℝ^{n} ⊃ V → M such that \overline{im π} is contained in the domain of some chart ϕ where ϕ ∘ π is a k-subanalytic modification.

3 k-subanalytic rectilinearization.

3.1 Cells with reducers.

Let E ⊂ ℝ_{x}^{n} × ℝ_{y}^{1} be a PSk_{n+1}-set.

A reducing system R on E is a system R = (θ, {ϕ_{1}, ..., ϕ_{s}}, a, b, {λ_{i}}_{i=1}^{r},

{λ̃_{k}}_{k=1}^{q}), where s, q, r ∈ ℕ^{∗}, θ, ϕ_{1}, ..., ϕ_{s}, a, b ∈ PSk_{n}, λ_{i}, λ̃_{k} ∈ k_{+}^{∗} and for

(x, y) ∈ E the following conditions hold:

y − θ(x) ≠ 0, functions y − θ(x), a(x), b(x) are of constant sign,
if θ(x) ≢ 0, then y ∼ θ(x),

the mapping ψ(x, y) = (ϕ_{1}(x), ..., ϕ_{s}(x), |a(x)·(y−θ(x))|^{λ_{1}}, ..., |a(x)·(y−θ(x))|^{λ_{r}},

|\frac{b(x)}{y − θ(x)}|λ̃_{1}, ..., |\frac{b(x)}{y − θ(x)}|λ̃_{q}) is bounded.

The mapping ψ(x, y) is called the reducing morphism of R.

A finite family of functions {f_{ν}}_{ν} ⊂ PSk_{n+1} is R-reducible on E, when every function f_{ν} has on E the form (called R-reduction)

(⋆) f_{ν}(x, y) = |y − θ(x)|^{γ_{ν}} · A_{ν}(x) · V_{ν}(ψ(x, y)),

where γ_{ν} ∈ k, A_{ν} ∈ PSk_{n}, V_{ν} is an analytic, nonvanishing function of constant sign on some neighbourhood of the set \overline{ψ(E)} ⊂ ℝ^{s+r+q} and for every ν, A_{ν}(x) = 0 or A_{ν}(x) ≠ 0 holds on E. (Thus also f_{ν} = 0 or f_{ν} ≠ 0 on E.)

A reducer R on K, where K is a PSk-cell in ℝ^{n+1}, is a sequence R = (R^{1}, ..., R^{n+1}), where R^{j} is a reducing systemem on K^{j} ⊂ ℝ^{j}.

For a finite family of functions {f_{ν}}_{ν} ⊂ PSk_{n+1} and a reducer R on K, where

R^{j} = (θ_{j}, {ϕ_{j,1}, ..., ϕ_{j,s_{j}}}, a_{j}, b_{j}, {λ_{j,i}}_{i=1}^{r_{j}}, {λ̃_{j,k}}_{k=1}^{q_{j}}), j = 1, ..., n + 1

we say that it is R-reducible on K, when is R^{n+1}-reducible on K:

f_{ν}(x, y) = |y − θ_{n+1}(x)|^{γ_{ν}} · A_{ν}(x)·
·V_{ν}(ϕ_{n+1,1}(x), ..., ϕ_{n+1,s_{n+1}}(x), |a_{n+1}(x) · (y − θ_{n+1}(x))|^{λ_{n+1,1}}, ...,

|a_{n+1}(x) · (y − θ_{n+1}(x))|^{λ_{n+1,r}}, |\frac{b_{n+1}(x)}{y − θ_{n+1}(x)}|λ̃_{n+1,1}, ..., |\frac{b_{n+1}(x)}{y − θ_{n+1}(x)}|λ̃_{n+1,q}),

and the family {θ_{n+1}, ϕ_{n+1,1}, ..., ϕ_{n+1,s_{n+1}}, a_{n+1}, b_{n+1}} ∪ {A_{ν}}_{ν} is (R^{1}, ..., R^{n})- reducible on K^{n}. (Constant functions on ℝ^{0} are reducible.)

Remark. If f ∈ PSk_{n+1}, λ ∈ k, and f is R-reducible on K, then |f|^{λ} is R-reducible on K.

A cell with a reducer in ℝ^{n+1} is a pair (K, R), where K is a cell in ℝ^{n+1}, a R is reducer on K such that for j = 1, ..., n the family {t_{j+1,1}, t_{j+1,2}} is (R^{1}, ..., R^{j})-reducible on K^{j}. We say that a finite family {f_{ν}}_{ν} ⊂ PSk_{n+1} is reducible on (K, R), when it is R-reducible on K.

The following proposition gives a kind of ”partial desingularization”.

Proposition 3.1.1 For a finite family {f_{ν}}_{ν} ⊂ PSk_{n+1} reducible on a cell with a reducer (K, R) in ℝ^{n+1}, there exist: an open neighbourhood U of the cell K and a k-subanalytic and analytic isomorphism Ω : U ∋ z ↦ z̃ ∈ Ũ of open sets in ℝ^{n+1} such that Ũ ⊂ (0, +∞)^{n+1} and for every ν

(f_{ν} ∘ Ω^{−1}| Ω(K)) (z̃_{1}, ..., z̃_{n+1}) = |z̃|^{γ_{ν}} · h_{ν}(z̃),

where γ_{ν} ∈ k^{n+1}, h_{ν} are zeroes or analytic and k-subanalytic units on Ũ. Thus {f_{ν}| K} are restrictions of analytic functions on U.

Proof. We will use induction on n. On K we have reductions with a common reducing system:

f_{ν}(x, y) = |y − θ(x)|^{γ_{ν}} · A_{ν}(x) · V_{ν}(ψ(x, y)),

where

ψ(x, y) = (ϕ_{1}(x), ..., ϕ_{s}(x), |a(x) · (y − θ(x))|^{λ_{1}}, ..., |a(x) · (y − θ(x))|^{λ_{r}},

|\frac{b(x)}{y − θ(x)}|λ̃_{1}, ..., |\frac{b(x)}{y − θ(x)}|λ̃_{q}) ,

γ_{ν} ∈ k, θ, A_{ν}, ϕ_{j}, a, b ∈ PSk_{n}, V_{ν} are analytic units on some common open neighbourhood W of the set \overline{ψ(K)} ⊂ ℝ^{s+r+q}. We can assume that they are k-subanalytic.

Let {g_{μ}}_{μ} = {θ, ϕ_{1}, ..., ϕ_{s}, a, b} ∪ {A_{ν}}_{ν}. There exist: an open neighbourhood U′ of the cell K′ and an analytic and k-subanalytic isomorphism Ω′ : U′ → Ũ′ ⊂ (0, +∞)^{n} such that for every μ

(g_{μ} ∘ (Ω′)^{−1}| Ω′(K′)) (x̃) = |x̃|^{δ_{μ}}k_{μ}(x̃) , where δ_{μ} ∈ k^{n}

and k_{μ} is a zero or an analytic and k-subanalytic unit on Ũ′. (For n = 0 it is obvious; for n > 0 we use the induction assumption for K′ and the family {g_{μ}}_{μ}.)

Let

g̃_{μ} : Ũ′ ∋ x̃ ↦ |x̃|^{δ_{μ}}k_{μ}(x̃) ∈ ℝ

and

Z = \{(x, y) ∈ U′ × ℝ| sgn(y − θ(x)) = sgn(\frac{1}{2}(t_{1} + t_{2})(x) − θ(x))\}.

Moreover, let Z̃ = {(x̃, y)|(x, y) ∈ Z},

τ : Z̃ ∋ (x̃, y) ↦ (x̃, ỹ) = (x̃, |y − θ̃(x̃)|) ∈ Ũ′ × (0, +∞),
ψ̃(x̃, ỹ) = (ϕ̃_{1}(x̃), ..., ϕ̃_{s}(x̃), |ã(x̃) · ỹ|^{λ_{1}}, ...,

|ã(x̃) · ỹ|^{λ_{r}}, |\frac{b̃(x̃)}{ỹ}|λ̃_{1}, ..., |\frac{b̃(x̃)}{ỹ}|λ̃_{q}) .

Notice that τ̃ is an analytic and k-subanalytic isomorphism, and ψ̃ is an analytic and k-subanalytic mapping on Ũ′ × (0, +∞).

Let us set Ũ = ψ̃^{−1}(W),

U = (τ ∘ (Ω′ × id_{ℝ}))^{−1}(Ũ) and Ω = τ ∘ (Ω′ × id_{ℝ})| U.

Notice that ψ|_{K} = ψ̃ ∘ Ω|_{K}. Let Ã_{ν}(x̃) = |x̃|^{ω_{ν}}h_{ν}(x̃).

Thus every function

(f_{ν} ∘ Ω^{−1}| Ω(K)) (x̃, ỹ) = |ỹ|^{γ_{ν}}|x̃|^{ω_{ν}}h_{ν}(x̃)(V_{ν} ∘ ψ̃)(x̃, ỹ),

is a restriction of an analytic and k-subanalytic functiom on Ũ ⊂ (0, +∞)^{n+1} of desired form. □

Corollary 3.1.2 If (K, R) is a PSk-cell with a reducer, then K is an analytic cell.

The preparation theorem for k-subanalytic functions can be stated as follows:

Theorem 3.1.3 (Lion-Rolin [5]) For a finite family of functions {f_{ν}}_{ν} ⊂ PSk_{n+1}, there exists a finite family {(C_{α}, R_{α})}_{α}, where:

1) {C_{α}}_{α} is a decomposition of ℝ^{n+1} into PSk_{n}-cylinders (we can require that cylinders to be compatible with a given PSk_{n+1}- set adding the characteristing function of this set to the family of functions).

2) R_{α} is a reducing system on C_{α}.

3) The family {f_{ν}}_{ν} is R_{α}-reducible on C_{α}.

Corollary 3.1.4 Let E be PSk_{n+1}-set, and {f_{ν}}_{ν} a finite family of functions from PSk_{n+1}. Then there exists a finite family of cells with reducers {(K_{α}, R_{α})}_{α} such that {K_{α}}_{α} form a decomposition of E and the family {f_{ν}}_{ν} is reducible on every (K_{α}, R_{α}).

If the family {f_{ν}}_{ν} is already R-reducible on E with some reducing system R, then we can additionally get R = R_{α}^{n+1} for every α.

Additionally, for a given finite family of PSk_{n+1}-sets, we can get compatibility of cells with this family of sets.

Remark. Assume that E is a decomposition of the set E into a finite family of PSk_{n+1}-sets. If Corollary 3.1.4 holds for every element of E (the family of functions remains unchanged), then it holds also for E.

The proof of Corollary 3.1.4 is left to the reader. □

Corollary 3.1.5 For every finite family of PSk_{n}-sets there exists a decomposition of ℝ^{n} into a finite family of analytic PSk_{n−1}-cells which are compatible with given sets.

Remark. Thus the family (Tarski system) of PSk-sets admits analytic cell decomposition and (finite) analytic stratifications (see also the proofs of D.12, D.16, and D.18 in [3] ).

3.2 Lemmas.

Let p be any natural number.

We say that (E, {f_{ν}}_{ν}) is a (D^{P})-pair, where E is a compact PSk_{n+1}-set, and {f_{ν}}_{ν} ⊂ PSk_{n+1} is a finite family of bounded functions on E, if:

there exists a finite family of pairs (π_{α}, I_{α}), where π_{α} : N_{α} → ℝ^{n+1} (N_{α} open in ℝ^{n+1}) is a k-subanalytic modification of class C^{p}, I_{α} is a compact box in ℝ_{+}^{n+1} ∩ N_{α}, such that ⋃_{α} π_{α}(I_{α}) = E and for all α, ν the function f_{ν} ∘ π_{α}| reg I_{α} is a restriction of a function f̃_{ν,α}, which is zero or a k-normal crossing of class C^{p} on N_{α}.

Remarks.

a) Boxes I_{α} do not have to be full-dimensional.

b) If each (E_{i}, {f_{ν}}_{ν}) is a (D^{p})-pair (i = 1, ..., k.) and E = ⋃_{i=1}^{k} E_{i}, then (E, {f_{ν}}_{ν}) is a (D^{p})-pair.

c) By Remark 2 from 2.4, it suffices that functions f̃_{ν,α} be quotients of k-normal crossings or zeroes.

d) If f_{ν_{0}} is continuous on E, then, for every α, f̃_{ν_{0},α}|I_{α} = f_{ν_{0}} ∘ π_{α}|I_{α}.

Lemma 3.2.1 If n = 0, then every pair (E, {f_{ν}}_{ν}) is a (D^{p})-pair.

Proof. By the Lion-Rolin preparation theorem and Remark b), we can assume that E is a point or a compact interval and {f_{ν}}_{ν} is reducible on reg E. For y ∈ reg E, where E = [t_{1}, t_{2}],

f_{ν}(y) = |y − θ|^{γ_{ν}} · A_{ν} · V_{ν}(|y − θ|),

where γ_{ν} ∈ k, λ_{i}, λ̃_{j} ∈ k_{+}^{∗}, a, b, A_{ν} ∈ ℝ, V_{ν} is a k-analytic unit on a ksubanalytic neighbourhood of a set \overline{ψ(E)} ⊂ ℝ and ψ : ℝ ∋ y ↦ |y − θ| ∈ ℝ. Moreover, θ ∉ reg E.

Let

π : ℝ ∋ y_{1} ↦ ϵ · |y_{1}|^{l} + θ ∈ ℝ,

where ϵ = sgn (\frac{1}{2}(t_{1} + t_{2}) − θ) and l is some number from 2ℤ which will be defined later. The mapping π is a composition of a k-power function and an affine isomorphism.

Let I = ψ̃(E), with ψ̃(y) = |ψ(y)|^{\frac{1}{l}}, and let us take an open k-subanalytic W ⊂ ℝ such that every Ṽ_{ν} is defined on W and I ⊂ W.

We can take the family {((π|W), I)}:

π(I) = E and f_{ν} ∘ π| _{reg I} is a restriction of a function

y_{1} ↦ |y_{1}|^{l·γ_{ν}} · A_{ν} · Ṽ_{ν}(|y_{1}|^{l}),

which is zero or a k-normal crossing of class C^{p} on W (see Remark c)) for sufficiently large l. The mapping π|W is analytic (even polynomial). □

Lemma 3.2.2 If E is fat (i.e. E = \overline{int E}), then (E, \{f_{ν}\}_{ν}) is a (D^{p})-pair with full-dimensional boxes I_{α}.

Proof. We use induction on n. The case n = 0 was proved in Lemma 3.2.1. Let n > 0 and let us assume that the Lemma holds for n − 1.

By Corollary 3.1.4 and Remark b), we can assume that E is a closure of a full-dimensional cell on which the family {f_{ν}}_{ν} is reducible with

R = (θ(x), {ϕ_{1}(x), ..., ϕ_{s}(x)}, a(x), b(x), {λ_{i}}_{i=1}^{r}, {λ̃_{k}}_{k=1}^{q}),

that is, it has an R-reduction on E of the form

f_{ν}(x, y) = |y − θ(x)|^{γ_{ν}}A_{ν}(x)V_{ν}(ψ(x, y)), where

ψ(x, y) = (ϕ_{1}(x), ..., ϕ_{s}(x), |a(x) · (y − θ(x))|^{λ_{1}}, ..., |a(x) · (y − θ(x))|^{λ_{r}},

|\frac{b(x)}{y − θ(x)}|λ̃_{1}, ..., |\frac{b(x)}{y − θ(x)}|λ̃_{q}).

Extracting some bounded functions as new functions ϕ_{i}(x), we can assume that

0 ≠ a(x) = \frac{1}{max\{|t_{1}(x) − θ(x)|, |t_{2}(x) − θ(x)|\}}

b(x) = min{|t_{1}(x) − θ(x)|, |t_{2}(x) − θ(x)|},

where t_{1} and t_{2} are functions generating the cylinder K. (We can assume that the number of functions ϕ_{i}(x) remains unchanged.)

By Corollary 3.1.4, there exists a finite family of cells with reducers {(K_{ξ}, R_{ξ})}_{ξ∈Ξ} compatible with sets

F_{ν} = {(x, y) ∈ ℝ^{n+1} : |A_{ν}(x)| < 1}

such that {K_{ξ}}_{ξ∈Ξ} is a decomposition of E and for every ξ ∈ Ξ family {f_{ν}}_{ν} is reducible on (K_{ξ}, R_{ξ}), where R_{ξ}^{n+1} = R and functions A_{ν}(x) remain unchanged.

By Remark b), it suffices to prove the Lemma for the closures of fulldimensional cells \overline{K_{ξ}}. Fix such ξ (we will drop this index).

Let

A′_{ν}(x) = \{ \begin{matrix}A_{ν}(x), & K ⊂ F_{ν} \\ \frac{1}{A_{ν}(x)}, & K ∩ F_{ν} = ∅. \end{matrix}

Denote by {g_{μ}}_{μ} the family

{a′(x), b(x), b′(x), θ(x), ϕ_{1}(x), ..., ϕ_{s}(x)} ∪ {A′_{ν}(x)}_{ν}

of bounded PSk_{n}-functions on the basis K′ of the cell K, where a′(x) = \frac{1}{a(x)} and b′(x) = a′(x) − b(x).

We use the induction assumption for (\overline{K′}, \{g_{μ}\}_{μ}).

There exists a finite family of pairs (Φ_{β}, J_{β}), where Φ_{β} : M_{β} → ℝ^{n} (M_{β} open in ℝ^{n}) is a k-subanalytic modification of class C^{p}, a J_{β} is a fulldimensional box in ℝ_{+}^{n} ∩ M_{β}, such that ⋃_{β} Φ_{β}(J_{β}) = \overline{K′} and for all β, μ the function g_{μ} ∘ Φ_{β}| reg J_{β} is a restriction of a function g̃_{μ,β}, which is zero or a k-normal crossing of class C^{p} on M_{β}.

The mappings Ψ_{β} = Φ_{β} × id_{ℝ} are k-subanalytic modifications of class C^{p}.

Let L_{β} = {(u, y) ∈ int J_{β} × ℝ| (t_{1} ∘ Φ_{β})(u) < y < (t_{2} ∘ Φ_{β})(u)}. Then ⋃_{β} Ψ_{β}(\overline{L_{β}}) = \overline{K}, so it suffices to prove the Lemma for each pair (\overline{L_{β}}, \{f_{ν} ∘ Ψ_{β}}_{ν}). Fix β (we will drop this index).

Let J = [ζ_{1}, η_{1}] × ... × [ζ_{n}, η_{n}] and J = {j|ζ_{j} = 0}, k_{+}^{J} = {γ ∈ k_{+}^{n}| γ_{j} ≠ 0 ⇒ j ∈ J }. Set

b̃(u) = t_{1}(u)|u|^{c_{1}},

ã′(u) = t_{2}(u)|u|^{c_{2}},
b̃′(u) = t_{3}(u)|u|^{c_{3}},

where c_{1}, c_{2}, c_{3} ∈ k_{+}^{J}, functions t_{1}, t_{2}, t_{3} are k-analytic positive units on M, where t_{1} can also be zero (in this case c_{1} = c_{2}).

Set

C^{1} = {(u, y_{1}) ∈ int J × ℝ_{+}| b̃(u) < y_{1} < ã′(u)}
and π : M × ℝ ∋ (u, y_{1}) ↦ (u, ϵ · y_{1} + θ̃(u)) ∈ M × ℝ,

where ϵ = sgn (\frac{1}{2}(t_{1}(x) + t_{2}(x)) − θ(x)) for x ∈ K (does not depend on x). Then π(\overline{C^{1}}) = \overline{L} and f̂_{ν}(u, y_{1}) = (f_{ν} ∘ Ψ ∘ π| C^{1})(u, y_{1}) =

= |y_{1}|^{γ_{ν}} Ã_{ν}(u)V_{ν}(ϕ̃_{1}(u), ..., ϕ̃_{s}(u), |\frac{1}{ã′} (x) · (y − θ̃(x))|^{λ_{1}}, ..., |\frac{1}{ã′} (x) · (y − θ̃(x))|^{λ_{r}},

|\frac{b̃(x)}{y − θ̃(x)}|λ̃_{1}, ..., |\frac{b̃(x)}{y − θ̃(x)}|λ̃_{q}),

where

Ã_{ν} = \{ \begin{matrix}Ã′_{ν} & when A′_{ν} = A_{ν} \\ \frac{1}{Ã′_{ν}} & when A′_{ν} = \frac{1}{A_{ν}} \end{matrix} .

By the assumption, Ã_{ν}(u) = |u|^{δ_{ν}}h_{ν}(u) z δ_{ν} ∈ k^{J} and h_{ν} = 0 or h_{ν} are k-analytic units on M.

Extracting the units \frac{1}{ã′} , b̃ as new functions ϕ̃_{i} and joining h_{ν} with V_{ν}, we get

f̂_{ν}(u, y_{1}) = |y_{1}|^{γ_{ν}}|u|^{δ_{ν}} V̂_{ν}(u, \frac{y_{1}}{|u|^{c_{2}}} , \frac{|u|^{c_{1}}}{y_{1}} ),

where V̂_{ν} are zeroes or k-analytic units.

By boudedness od the function \frac{b̃(u)}{ã′(u)} on J, c_{2} ≤ c_{1}.

Let

C^{2} = {(u, y_{2}) ∈ int J × ℝ| t_{1}(u) |u|^{c_{1}−c_{2}} < y_{2} < t_{2}(u)}

and

R_{1} : ℝ^{n+1} ∋ (u, y_{2}) ↦ (u, y_{2} · |u|^{c_{2}}) ∈ ℝ^{n+1}

(a composition of blows-up and k-power functions). Then R_{1}(\overline{C^{2}}) = \overline{C^{1}} and

(f̂_{ν} ∘ R_{1}| C^{2})(u, y_{2}) = |y_{2}|^{γ_{ν}}|u|^{ω_{ν}} V̂_{ν}(u, y_{2}, \frac{|u|^{c_{1}−c_{2}}}{y_{2}} ),

where ω_{ν} = δ_{ν} + γ_{ν} · c_{2}.

Case 1: b̃(u) ≠ 0, c_{1} = c_{2}.

Then y_{2} is a unit on C^{2} and t_{2} − t_{1} = t_{3}|u|^{c_{3}−c_{2}} (c_{3} ≥ c_{2}).

Let us introduce the mappings

T : M × ℝ ∋ (u, y_{3}) ↦ (u, y_{3} · t_{3}(u) + t_{1}(u)) ∈ M × ℝ
R_{2} : ℝ^{n+1} ∋ (u, y_{4}) ↦ (u, y_{4} · |u|^{c_{3}−c_{2}}) ∈ ℝ^{n+1}

and cells

C^{3} = {(u, y_{3}) ∈ int J × ℝ| 0 < y_{3} < |u|^{c_{3}−c_{2}}}
C^{4} = {(u, y_{4}) ∈ int J × ℝ| 0 < y_{4} < 1}.

Then T (\overline{C^{3}}) = \overline{C^{2}}, R_{2}(\overline{C^{4}}) = \overline{C^{3}} and

(f̂_{ν} ∘ R_{1} ∘ T ∘ R_{2}| C^{4})(u, y_{4}) = |u|^{ω_{ν}}|y_{2}(u, y_{4})|^{γ_{ν}} V̂_{ν}(u, y_{2}(u, y_{4}), \frac{1}{y_{2}} (u, y_{4})).

Each of functions (V̂_{ν}(u, y_{2}, \frac{1}{y_{2}} ))(u, y_{4}) is zero or a k-analytic unit on a common open k-subanalytic neighbourhood Ŵ of a set \overline{C^{4}}. We substitute a kpower function P with sufficiently large exponents and set N_{α} = P^{−1}((M × ℝ) ∩ Ŵ), π_{α} = Ψ ∘ π ∘ R_{1} ∘ T ∘ R_{2} ∘ P | N_{α} and I_{α} = P^{−1}(\overline{C^{4}}) ∩ ℝ_{+}^{n+1} Both the k-subanalytic modification π_{α}, and the desingularized functions are of class C^{p}.

Case 2: b̃(u) = 0.

Then V̂_{ν} does not depend on the last variable. Applying the mapping

T′ : M × ℝ ∋ (u, y_{3}) ↦ (u, y_{3} · t_{2}(u)) ∈ M × ℝ

we get

T′(\overline{C^{3}}) = \overline{C^{2}} and (f̂_{ν} ∘ R_{1} ∘ T′)(u, y_{3}) = |y_{3}|^{γ_{ν}}|u|^{ω_{ν}}|t_{2}(u)|^{γ_{ν}} ˆV̂_{ν}(u, y_{3})

for (u, y_{3}) ∈ C^{3} = ( int J)×(0, 1), where ˆ V̂_{ν} are zeroes or k-analityc units on a common open k-subanalytic neighbourhood Ŵ of a set \overline{C^{3}}. We substitute a k-power function P with sufficiently large exponents and set I_{α} = P^{−1}(\overline{C^{3}})∩ ℝ_{+}^{n+1}, N_{α} = P^{−1}((M × ℝ) ∩ Ŵ), and π_{α} = Ψ ∘ π ∘ R_{1} ∘ T′ ∘ P | N_{α}. Both the k-subanalytic modification π_{α}, and the desingularized functions are of class C^{p}.

Case 3: c_{1} ≠ c_{2} and b̃(u) ≠ 0.

We put

C^{3} = \{(u, y_{3}) ∈ int J × ℝ : |u|^{c_{1}−c_{2}} < y_{3} < \frac{t_{2}}{t_{1}} (u)\} and

T′′ : M × ℝ ∋ (u, y_{3}) ↦ (u, y_{3} · t_{1}(u)) ∈ M × ℝ.

Then

T′′(\overline{C^{3}}) = \overline{C^{2}} and (f̂_{ν} ∘ R_{1} ∘ T′′| C^{3})(u, y_{3}) = |y_{3}|^{γ_{ν}}|u|^{ω_{ν}} ˆV̂_{ν}(u, y_{3}, \frac{|u|^{c_{1}−c_{2}}}{y_{3}} ).

Making, if necessary, a decomposition of the cell C^{3}, we can assume that |η|^{c_{1}−c_{2}} < \frac{t_{2}}{t_{1}} (u) on C^{3} or J = ∅, reducing the situation to Case 1.

Let 0 = κ_{0} < κ_{1} < ... < κ_{k} = c_{1} − c_{2}, where κ_{j} ∈ k_{+}^{J} and κ_{j} − κ_{j−1} = ρ_{j} · e_{i(j)}, ρ_{j} ∈ k_{+}^{∗}. We set

C_{0}^{3} = \{(u, y_{3}) ∈ int J × ℝ : |η|^{κ_{k}} < y_{3} < \frac{t_{2}}{t_{1}} (u)\}

C_{j}^{3} = {(u, y_{3}) ∈ int J × ℝ : |η|^{κ_{k}−κ_{j}}|u|^{κ_{j}} < y_{3} < |η|^{κ_{k}−κ_{j−1}}|u|^{κ_{j−1}}, j = 1, ..., k.

It suffices to prove the Lemma for each pair (\overline{C_{j}^{3}}, \{f̂_{ν} ∘R_{1} ∘T′′\}_{ν}), j = 0, ..., k. For j = 0, Case 1 holds. For j = 1, ..., k, we put

C_{j}^{4} = {(u, y_{4}) ∈ int J × ℝ : |η|^{κ_{k}−κ_{j}}|u_{i(j)}|^{ρ_{j}} < y_{4} < |η|^{κ_{k}−κ_{j}}|η_{i(j)}|^{ρ_{j}}}

R_{2} : ℝ^{n+1} ∋ (u, y_{4}) ↦ (u, y_{4} · |u|^{κ_{j−1}}) ∈ ℝ^{n+1}.

Then R^{2}(\overline{C_{j}^{4}}) = \overline{C_{j}^{3}} and

(f̂_{ν} ∘ R_{1} ∘ T′′ ∘ R_{2}| C_{j}^{4})(u, y_{4}) = |y_{4}|^{γ_{ν}}|u|^{ω′_{ν}}H_{ν}(u, y_{4}, \frac{|u_{i(j)}|^{ρ_{j}}}{y_{4}} ),

where ων^{′} = ω_{ν} +γ_{ν} ·|u|^{κ_{j−1}}, H_{ν} is zero or a k-analytic unit in a neighbourhood W of the set \overline{ψ̂(C_{j}^{4})}, where ψ̂(u, y_{4}) = (u, y_{4}, \frac{|u_{i(j)}|^{ρ_{j}}}{y_{4}} ). Moreover let

C_{j}^{5} = {(u, y_{5}) ∈ int J × ℝ : |u_{i(j)}|^{ρ_{j}} < y_{5} < |η_{i(j)}|^{ρ_{j}}}
T′′′ : ℝ^{n+1} ∋ (u, y_{5}) ↦ (u, y_{5} · |η|^{κ_{k}−κ_{j}}) ∈ ℝ^{n+1}.

Then T′′′(\overline{C_{j}^{5}}) = \overline{C_{j}^{4}}. We can assume that i(j) = n. Set J′ = [ζ_{1}, η_{1}] × ... × [ζ_{n−1}, η_{n−1}] and

Ĉ_{j}^{5} = {(û, y_{5}) ∈ int J′ × ℝ^{2}| 0 < û_{n} < 1, 0 < y_{5} < η_{n}}
R̂ : ℝ^{n+1} ∋ (û, y_{5}) ↦ (û_{1}, ..., û_{n−1}, |û_{n} · y_{5}|^{\frac{1}{ρ_{j}}} , y_{5}) ∈ ℝ^{n+1}.

Then R̂(\overline{Ĉ_{j}^{5}}) = \overline{C_{j}^{5}} and

(f̂_{ν}∘R_{1}∘T′′∘R_{2}∘T′′′∘R̂|Ĉ_{j}^{5})(û, y_{5}) = |y_{5}|^{l′_{ν}}|û|^{ω′_{ν}}H_{ν}^{⋆}(û_{1}, ..., û_{n−1}, |û_{n}·y_{5}|^{\frac{1}{ρ_{j}}} , y_{5}, û_{n}),

where lν^{′} = γ_{ν} + (ων^{′})_{n}. Thus the above functions are of the form |ẑ|^{σ_{ν}} Ĥ_{ν}(ẑ), where ẑ = (û, y_{5}), σ_{ν} ∈ k^{n+1}, and every Ĥ_{ν} is zero or a k-analytic unit in a common open k-subanalytic neighbourhood Ŵ of the set \overline{Ĉ_{j}^{5}}. We substitute a

k-power function P with sufficiently large exponents and set I_{α} = P^{−1}(\overline{Ĉ_{j}^{5}})∩ ℝ_{+}^{n+1}, N_{α} = P^{−1}(Ŵ ∩(R̂)^{−1}(M ×ℝ)) and π_{α} = f̂_{ν} ∘R_{1}∘T′′∘R_{2}∘T′′′∘R̂∘P |N_{α}. Both the k-subanalytic modification π_{α}, and the desingularized functions are of class C^{p}. □

Lemma 3.2.3 For a bounded cylinder C ⊂ ℝ^{n+1} with basis B ⊂ ℝ^{n} and generating functions t_{1}, t_{2} ∈ PSk_{n}, if the pair (\overline{B}, \{t_{1}, t_{2}\}) is a (D^{p})-pair, then for any finite family \{f_{ν}\}_{ν} ⊂ PSk_{n+1} of functions bounded on \overline{C}, the pair (\overline{C}, \{f_{ν}\}_{ν}) is a (D^{p})-pair.

Proof. We can assume that {t_{1}, t_{2}} is reducible on (B, R) with some reducer R. By the assumption, there exists a finite family of pairs (Φ_{β}, J_{β}), where Φ_{β} : M_{β} → ℝ^{n} is a k-subanalytic modification of class C^{p}, J_{β} is a box in ℝ_{+}^{n} ∩ M_{β}, such that ⋃_{β} Φ_{β}(J_{β}) = \overline{B} and for each β the functions t_{i} ∘ Φ_{β}| reg J_{β} (i = 1, 2) are restrictions of functions t̃_{i,β} that are zeroes or k-normal crossings of class C^{p} on M_{β}.

Let Ψ_{β} = Φ_{β} × id_{ℝ}.

The box J_{β} is isomorphic to a fat box J̃_{β} in ℝ^{k} (0 ≤ k ≤ n) through the mapping L_{β} : ℝ^{k} ⊃ J̃_{β} → J_{β} ⊂ ℝ^{n}. Let L_{β,1} = L_{β} × id_{ℝ}. We can assume that J_{β} = {b} × J̃_{β}, with b ∈ ℝ_{+}^{n−k}.

Case 1. C is of the form {(x, y) ∈ B × ℝ| t_{1}(x) < y < t_{2}(x)}.

The proof of Case 1 is similar to the proof of Case 2.

Case 2. C is of the form {(x, y) ∈ B × ℝ| t_{1}(x) = y = t_{2}(x)}.

We define

C_{β} = {(u, y) ∈ reg J_{β} × ℝ| (t_{1} ∘ Φ_{β})(u) = y}
C̃_{β} = int J̃_{β} ⊂ ℝ^{k}.

Then ⋃_{β} Ψ_{β} ∘ L_{β,1} ∘ (id_{ℝ}_{k}, t̃_{1,β})(J̃_{β}) = \overline{C}.

We apply Lemma 3.2.2 do each pair (J̃_{β}, {f_{ν} ∘ Ψ_{β} ∘ L_{β,1} ∘ (id_{ℝ}_{k}, t̃)}_{ν}), where t̃(v) = t̃_{1,β}(b, v).

There exists a finite family of pairs (π̂α̂, Îα̂), where π̂α̂ : N̂α̂ → ℝ^{k} is a k-subanalytic modification of class C^{p}, Îα̂ is a fat box in ℝ_{+}^{k} ∩ N̂α̂, such that ⋃α̂ π̂α̂(Îα̂) = J̃_{β} and for every α̂ and ν the function f_{ν} ∘ Ψ_{β} ∘ L_{β,1} ∘ (id_{ℝ}_{k}, t̃) ∘ π̂α̂| int Îα̂ is a restriction of the function f̃_{ν,α̂}, which is zero or a k-normal crossing of class C^{p} on N̂α̂.

Let N_{α} = ℝ^{n−k} ×N̂α̂ × ℝ, I_{α} = {b}×Îα̂ ×{0}. The mapping π_{α} = Ψ_{β} ∘Γ∘ (id_{ℝ}_{n−k} × π̂α̂ × id_{ℝ}) : N_{α} → ℝ^{n+1}, where Γ(x_{1}, ..., x_{n+1}) = (x_{1}, ..., x_{n}, x_{n+1} + t̃(x_{n−k+1}, ..., x_{n})), is a k-subanalytic modification of class C^{p}.

Then ⋃_{α} π_{α}(I_{α}) = \overline{C} and each of functions

f_{ν} ∘ π_{α}| reg I_{α} = f_{ν} ∘ Ψ_{β} ∘ L_{β,1} ∘ (id_{ℝ}_{k}, t̃) ∘ π̂α̂ ∘ ρ′| reg I_{α},

where ρ′ : ℝ^{n+1} ∋ (w, v, y) ↦ v ∈ ℝ^{k}, is zero or a restriction of a k-normal crossing on N_{α}. This proves that (\overline{C}, \{f_{ν}\}_{ν}) is a (D^{p})-pair. □

Corollary 3.2.4 Each pair (E, {f_{ν}}_{ν}) is a (D^{p})-pair.

Proof. If E is the closure of an analytic cell, then it suffices to use inductively Lemma 3.2.3. For any E, we use Remark b) and Corollary 3.1.4.

□

Remark. We can additionally obtain π_{α} as combinations of k-normal crossings. Indeed, it suffices to use the fact that (I_{α}, {f_{ν}∘π_{α}}∪{π_{α,1}, ..., π_{α,n+1}}) is a (D^{p})-pair for all α. In this case we can treat all functions as globally defined extending them by zero.

3.3 Rectilinearization and uniformization theorems.

Theorem 3.3.1 (on rectilinearization in ℝ^{n} ) Let {E_{ν}}_{ν} be a finite family of k-subanalytic bounded sets in ℝ^{n} and let f_{ν} : E_{ν} → ℝ be a family of k-subanalytic continuous functions. Then there exists a finite family of mappings Φ_{α} : ℝ^{l(α)} → ℝ^{n} such that:

a) every Φ_{α} is a composition of an inclusion ℝ^{l(α)} → ℝ^{n} and global ksubanalytic modification of class C^{p}, which is a combination of local k-normal crossings,

b) there are compact boxes L_{α} ⊂ ℝ^{l(α)} such that ⋃_{α} Φ_{α}(L_{α}) is a neighbourhood of a set ⋃_{ν} \overline{E_{ν}} and each of \overline{E_{ν}} is a union of some Φ_{α}(L_{α})

c) each of sets Φ_{α}^{−1}(E_{ν}) is a boolean combination of hyperplanes H_{i}^{l} = {x ∈ ℝ^{l} : x_{i} = 0},

d) if Φ_{α}^{−1}(E_{ν}) is dense in ℝ^{l(α)}, then every f_{ν} ∘ Φ_{α} : Φ_{α}^{−1}(E_{ν}) → ℝ is a restriction of a function f_{ν}^{⋆} : ℝ^{l(α)} → ℙ, which is zero, a local k-normal crossing of class C^{p} or an inverse of a local k-normal crossing of class C^{p}.

Proof. Let F be a compact and k-subanalytic neighbourhood of a set ⋃_{ν} \overline{E_{ν}}. Each of sets E_{ν} is a boolean combination of k-subanalytic compact sets E_{ν}^{j}, j = 1, ..., r_{ν}. On F we consider a family of functions d_{ν}^{j} = dist(·, E_{ν}^{j}).

By Remark b) from 3.2, we can assume that for every ν: (a) |f_{ν}| ≤ 1 on E_{ν} or (b) |f_{ν}| ≤ 1 on E_{ν}. Let

f̂_{ν}(x) = \{ \begin{matrix}f_{ν}(x) & for x ∈ E_{ν}, when (a), \\ \frac{1}{f_{ν}(x)} & for x ∈ E_{ν}, when (b) and not (a), \\ 0 & when x ∉ E_{ν}. \end{matrix}

By Corollary 3.1.5 and the following Remark, there exists a stratification {S_{ξ}}_{ξ} of ℝ^{n} on analytic k-subanalytic cells compatible with the family of sets \{F \}∪\{E_{ν}^{j}\}. Each pair (\overline{S_{ξ}}, \{f̂_{ν}\}∪\{d_{ν}^{j}\}) with S_{ξ} being maximal cells for some E_{ν} or for F ^{1}, is a (D^{p})-pair.

We can assume that F = \overline{S_{ξ}} and drop the index ξ. There exists a finite family of pairs (ϕ_{β}, I_{β}), where ϕ_{β} : M_{β} → ℝ^{n} is a k-subanalytic modification of class C^{p} and a combination of k-normal crossings, I_{β} is a compact box in M_{β} ∩ ℝ_{+}^{n} such that ⋃_{β} ϕ_{β}(I_{β}) = \overline{S}, ⋃_{β} ϕ_{β}(reg I_{β}) ⊂ S and for any ν, β the function f̂_{ν} ∘ ϕ_{β}|reg I_{β} is a restriction of a function f̃_{ν,β} : M_{β} → ℝ, which is zero or a k-normal crossing of class C^{p}.

1i.e. open in some E_{ν} or in F

If S ⊄ E_{ν}, then, for every β, ϕ_{β}^{−1}(E_{ν}) ∩ reg I_{β} = ∅.

If S ⊂ E_{ν}, then by the continuity of functions f̂_{ν} on E_{ν},

f̂_{ν} ∘ ϕ_{β} = f̃_{ν,β} on ϕ_{β}^{−1}(E_{ν}) ∩ I_{β}.

Moreover,

d_{ν}^{j} ∘ ϕ_{β} = d̃_{ν,β}^{j} on I_{β}.

Thus each of sets Z_{j,ν,β} = (ϕ_{β}|I_{β})^{−1}(E_{ν}^{j}) = (d_{ν}^{j} ∘ ϕ_{β}|I_{β})^{−1}(0) is a union of closed walls of codimension at most one in I_{β}.

For each box I_{β}, we apply Lemma 2.3.1: there is a finite family of semialgebraic modifications m_{μ,β} : ℝ^{n} → ℝ^{n} being combinations of local k-normal crossings and a compact box L_{β} ⊂ ℝ^{dim I_{β}} such that ⋃_{μ} m_{μ,β}(ℝ^{dim I_{β}}) = ⋃_{μ} m_{μ,β}(L_{β}) = I_{β}.

Let α = (β, μ), L_{α} = L_{β}, l(α) = dim I_{β} and Φ_{α} = ϕ_{β} ∘m_{μ,β} ∘i_{l(α)} : ℝ^{l(α)} → ℝ^{n}.

The mapping Φ_{α} is a composition of an inclusion i_{l(α)} and a k-subanalytic modification ϕ_{β} ∘ m_{μ,β} of class C^{p}, which is a composition local k-normal crossings. Now, ⋃_{α} Φ_{α}(L_{α}) = F and Φ_{α}^{−1}(E_{ν}^{j}) = i_{l(α)}^{−1} ∘ m_{μ,β}^{−1}(Z_{j,ν,β}) is a union of hyperplanes H_{i}^{l} or the whole space ℝ^{l(α)}. Thus Φ_{α}^{−1}(E_{ν}) is a boolean combination of such hyperplanes.

If Φ_{α}^{−1}(E_{ν}) is dense, then f̂_{ν} ∘ Φ_{α} : Φ_{α}^{−1}(E_{ν}) → ℝ is a restriction of a function f̃_{ν,β} ∘ m_{μ,β} ∘ i_{l(α)} : ℝ^{l(α)} → ℝ, which is a local k-normal crossing of class C^{p} or zero. □

The above version of the rectilinearization theorem for sets bounded in ℝ^{n} can be moved to manifolds:

Theorem 3.3.2 (rectilinearization theorem) Let {E_{ν}}_{ν} be a finite family of local k-subanalytic sets on an analytyic manifold M of dimension n and let f_{ν} : E_{ν} → ℝ be a family of continuous and k-subanalytic functions in M. Then there exists a locally finite family of mappings Φ_{α} : ℝ^{l(α)} → M such that:

a) every Φ_{α} is a composition of an inclusion ℝ^{l(α)} → ℝ^{n} and a global k-subanalytic modification of class C^{p} into M,

b) there are compact boxes L_{α} ⊂ ℝ^{l(α)} such that ⋃_{α} Φ_{α}(L_{α}) = M and each of \overline{E_{ν}} is a union of some Φ_{α}(L_{α}),

c) each of sets Φ_{α}^{−1}(E_{ν}) is a boolean combination of hyperplanes H_{i}^{l} = {x ∈ ℝ^{l} : x_{i} = 0},

d) if Φ_{α}^{−1}(E_{ν}) is dense in ℝ^{l(α)}, then every f_{ν} ∘ Φ_{α} : Φ_{α}^{−1}(E_{ν}) → ℝ is a restriction of the zero function, a local k-normal crossing of class C^{p} or an inverse of a local k-normal crossing of class C^{p}. □

Corollary 3.3.3 (uniformization theorem) Let E be a closed and locally k-subanalytic set of dimension k on an analytic manifold M. Then there exists an analytic manifold N of dimension k and a proper mapping f : N → M which is locally k-subanalytic of class C^{p} and such that f(N) = E.

Proof. We substitute the projections of k-dimensional toruses onto the boxes L_{α} ⊂ ℝ^{l(α)} in the mappings Φ_{α} from the rectilinearization theorem.

□

Remark. We can mix uniformization with rectilinearization of functions.

3.4 Subanalytic case.

In the subanalytic case, we can make parallel reasonings and obtain versions of classical desingularization theorems. It is due to the fact that rational exponents appearing in the Parusiński-Lion-Rolin formula always have a common divisor.

Theorem 3.4.1 (rectilinearization theorem) (cf. Desingularization II in [4], thm. 0.2 in [1], thm. 2.6 and thm. 2.7 in [7]) Let {E_{ν}}_{ν} be a finite family sets of subanalytic sets on an analytic manifold M of dimension n and let f_{ν} : E_{ν} → ℝ be a family of continuous, globally subanalytic functions on M. Then there exists a locally finite family of mappings Φ_{α} : ℝ^{l(α)} → M such that:

a) every Φ_{α} is a composition of an inclusion ℝ^{l(α)} → ℝ^{n} and a global subanalytic modification into M,

b) there exist compact boxes L_{α} ⊂ ℝ^{l(α)} such that ⋃_{α} Φ_{α}(L_{α}) = M and each of \overline{E_{ν}} is a union of some Φ_{α}(L_{α}),

c) each of sets Φ_{α}^{−1}(E_{ν}) is a boolean combination of hyperplanes H_{i}^{l} = {x ∈ ℝ^{l} : x_{i} = 0},

d) if Φ_{α}^{−1}(E_{ν}) is dense in ℝ^{l(α)}, then every f_{ν} ∘ Φ_{α} : Φ_{α}^{−1}(E_{ν}) → ℝ is a restriction of a function f_{ν}^{⋆} : ℝ^{l(α)} → ℙ, which is zero, a local normal crossing or an inverse of a local normal crossing. □

Corollary 3.4.2 (uniformization theorem) (cf. Desingularization I in [4], thm. 0.1 in [1]) Let E be a closed and subanalytic set of dimension k in an analytic manifold M. Then there exist: an analytic manifold N of dimension k and an analytic proper mapping f : N → M such that f(N) = E. □

INSTYTUT MATEMATYKI POLITECHNIKA KRAKOWSKA WARSZAWSKA 24 31-155 KRAKÓW POLAND E-mail: pupiekos@cyf-kr.edu.pl

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