Addendum to the paper

”Decomposition into special cubes and its application to quasi-subanalytic geometry”

Krzysztof Jan Nowak

IMUJ PREPRINT 2009/06

In paper [7], we demonstrate how to achieve the model completeness and o-minimality of the real field with restricted quasianalytic functions (result due to Rolin–Speissegger–Wilkie [13]) by means of a technique of decomposition into special cubes; see [8, 9, 10, 11] for other applications of this method. Therein we asked, inter alia, whether, given a polynomially bounded o-minimal expansion R of the real field, the structure generated by global smooth R-definable functions is model complete. We should note that this follows immediately from Wilkie’s complement theorem [14] (see also [12, 6]). In this Addendum, we also wish to indicate that Gabrielov’s proof [5] of the complement theorem can be adopted to the real field with restricted smooth R-definable functions.

Gabrielov’s approach relies on certain three preliminary lemmas. Below we state their quasianalytic versions, whose proofs can be repeated mutatis mutandis. Next, we shall outline our proof of the complement theorem based on those lemmas. Denote by Q_{n} the algebra of those R-definable functions that are smooth in the vicinity of the closed cube [0, 1]^{n}. The algebras Q_{n} give rise to the notions of Q-analytic, Q-semianalytic and Q-subanalytic subsets of the cubes [0, 1]^{n}, n ∈ ℕ.

AMS Classification: 14P15, 32B20, 26E10.

Key words: quasianalytic functions, quasi-subanalytic sets, complement theorem.

Lemma 1. Consider a Q-semianalytic subset E of [0, 1]^{n} of the form

E := {x ∈ [0, 1]^{n} : f_{1}(x) = … = f_{k}(x) = 0, g_{1}(x) > 0, …, g_{l}(x) > 0}

with f_{i}, g_{j} ∈ Q_{n}. Then the closure \overline{E} and frontier ∂E are Q-semianalytic too. Moreover, \overline{E} and ∂E can be described by functions which are polynomials in x, in the functions f_{i}, g_{j}, and in their (finitely many) partial derivatives.

Consequently, if F is a Q-subanalytic subset of [0, 1]^{m}, then so are its closure \overline{F} and frontier ∂F.

Remark. As an easy generalization, one can formulate the parametric version of the above lemma, in which the R-definable functions involved in the description depend smoothly on parameters.

By a Q-leaf we mean a set of the form

L := {x ∈ [0, 1]^{n} : f_{1}(x) = … = f_{k}(x) = 0, g_{1}(x) > 0, …, g_{l}(x) > 0},

where f_{i}, g_{j} ∈ Q_{n} and

\frac{∂(f_{1}, …, f_{k})}{∂(x_{i_{1}}, …, x_{i_{k}})}(x) ≠ 0 for some 1 ≤ i_{1} < … < i_{k} ≤ n and for all x ∈ L.

Lemma 2. Every Q-semianalytic subset E of [0, 1]^{n} is a finite union of Q-leaves.

The image of a Q-leaf L ⊂ [0, 1]^{n} under a projection π : ℝ^{n} ⟶ ℝ^{m}, n ≥ m, shall be called an immersed Q-leaf, if the restriction of π to L is an immersion. By combining Lemma 2 with the technique of fiber cutting (see e.g. [4, 5, 2, 3, 1, 7]), one can obtain

Lemma 3. Every Q-subanalytic subset F of [0, 1]^{m} is a finite union of immersed Q-leaves.

By a Q-cell we mean a cell given by smooth functions with Q-subanalytic graphs. Now we can readily outline our proof of the following main result wherefrom the complement theorem follows immediately.

Main Theorem. Consider Q-subanalytic subsets F_{1}, …, F_{r} of [0, 1]^{m}. Then there exists a Q-cell decomposition C of [0, 1]^{m} which is compatible with the sets F_{i}, i = 1, …, r.

We proceed by a double induction with respect to m and

d := max { dim F_{1}, …, dim F_{r} }.

The case m = 0 is trivial, and so take m > 0. Again, the case d = 0 is evident, and we may suppose d > 0. By virtue of Lemma 3, we can assume that F_{i} are immersed Q-leaves, i.e.

F_{i} = p(E_{i}), p : ℝ^{n} ⟶ ℝ^{m}, p(x_{1}, …, x_{m}) = (x_{1}, …, x_{n}),

for all i = 1, …, r. Denote by q : ℝ^{n} ⟶ ℝ^{m−1} and π : ℝ^{m} ⟶ ℝ^{m−1} the canonical projections onto the first m − 1 coordinates; obviously, π ∘ p = q. Put d_{i} := dim F_{i} = dim E_{i}, d_{i} ≤ d, i = 1, …, r, and

E′_{i} := {x ∈ E_{i} : rank q|E_{i} = d_{i}}, E′′_{i} := {x ∈ E_{i} : rank q|E_{i} = d_{i} − 1}.

Then E_{i} = E′_{i} ∪ E′′_{i}. Clearly, the restriction

res q : E′_{i} \ q^{−1}(q(∂E′_{i})) ⟶ q(E′_{i}) \ q(∂E′_{i})

is proper. Now observe that the set S of self-intersections of the image of res q is a Q-subanalytic subset of q(E′_{i}) as S ×\{0\} = \overline{V} ∩(q(E′_{i})×\{0\}), where

V := { (u_{1}, …, u_{m−1}, ϵ) ∈ (q(E′_{i}) \ q(∂E′_{i})) × [0, 1] :
∃ v = (v_{1}, …, v_{n−m+1}), w = (w_{1}, …, w_{n−m+1}) ∈ [0, 1]^{n−m+1} :
0 < |v − w| < ϵ, (u, v), (u, w) ∈ E′_{i} \ q^{−1}(q(∂E′_{i})) }.

Then T := S ∪ q(∂E′_{i}) is a Q-subanalytic set of dimension < d, and the restriction

res q : E′_{i} \ q^{−1}(T) ⟶ q(E′_{i}) \ T

is a topological covering, whence so is the restriction

res π : p(E′_{i}) \ π^{−1}(T) ⟶ q(E′_{i}) \ T.

Therefore, the set p(E′_{i}) is over any connected subset (in the sequel we shall consider a Q-cell) of q(E′_{i}) \ T is a finite union of the Q-subanalytic graphs of smooth functions.

Further, notice that, for each u ∈ q(E′′_{i}), the fiber (E′′_{i})_{u} := q^{−1}(u)∩E′′_{i} is a smooth Q-semianalytic arc, and the restriction of p to (E′′_{i})_{u} is an immersion of this fiber into {u} × ℝ_{x_{m}} whence the fiber (F_{i})_{u} is a finite union of open intervals. By virtue of the parametric version of Lemma 1, the sets

Z_{i} := ⋃_{u∈q(E′′_{i})}({u} × ∂p(E′′_{i})_{u}) ⊂ [0, 1]^{m}

are Q-subanalytic of dimension < d. By induction hypothesis, there exists a Q-cell decomposition {C_{p} : p = 1, …, s} of [0, 1]^{m} compatible with the sets Z_{i}, i = 1, …, r. Clearly, for each cell C_{p}, the sets

W_{i,p} := {u ∈ [0, 1]^{m−1} : (C_{p})_{u} ⊂ (E_{i})_{u}} ⊂ [0, 1]^{m−1}

are Q-subanalytic. Again by induction hypothesis, one can find a Q-cell decomposition C compatible with the sets

q(E′_{i}), q(∂E′_{i}), W_{i,p}, p(E′_{i}) ∩ π^{−1}q(∂E′_{i}) and Z_{i};

whereas the first three are subsets of [0, 1]^{m−1}, the last two are subsets of [0, 1]^{m} of dimension < d. Indeed, one must construct a Q-cell decomposition compatible with the subsets of [0, 1]^{m} under study, which are of dimension < d, and next refine the induced Q-cell decomposition of [0, 1]^{m−1} compatibly with the remaining subsets of [0, 1]^{m−1}.

What still remains to be done is to modify the Q-cell decomposition C, achieved in this fashion, as follows. As we have already seen, over each Q-cell C from the induced Q-cell decomposition of [0, 1]^{m−1} such that C ⊂ q(E′_{i}) but C ∩ q(∂E′_{i}) = ∅, i = 1, …, r, the set p(E′_{i}) is a finite union of the Q-subanalytic graphs of smooth functions. Again, one must modify C by partitioning its Q-cells by means of those Q-subanalytic graphs; this is, of course, linked with a successive refinement of the cube [0, 1]^{m−1}, which is possible due to induction hypothesis.

It is not difficult to check that eventually we attain a Q-cell decomposition C of [0, 1]^{m} compatible with the sets p(E′_{i}) and p(E′′_{i}), and a fortiori with the sets F_{i} := p(E_{i}) = p(E′_{i}) ∪ p(E′′_{i}). We leave the details for the reader.

Acknowledgements. This research was partially supported by Research Project No. N N201 372336 from the Polish Ministry of Science and Higher Education.

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Institute of Mathematics Jagiellonian University ul. S. Łojasiewicza 6 30-348 Kraków, Poland e-mail address: nowak@im.uj.edu.pl