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Sub-Pfaffian sets and a generalization of Wilkie’s theorem. (English) Zbl 1034.32008
This paper gives an analytic generalisation of Wilkie’s theorem on the model completeness of the theory of the real field with exponentiation. Let $$I=(-1,1)$$, $$I_+=(0,1)$$, $$I_-=(-1,0)$$ and for the canonical basis $$e_1,\dots,e_n$$ in $$\mathbb R^n \subset \oplus_{i=1}^{\infty}\mathbb R$$ and any increasing multi-index $$\alpha =(\alpha_1,\dots,\alpha_k)\in N_*^k$$ put $$I_{\alpha_i}=Ie_i$$, and $$I_{\alpha}= I_{\alpha_1}+ \cdots+I_{\alpha_k}$$. If $$\beta$$ is another multi-index such that $$\{\beta_1,\dots, \beta_s\} \subset \{\alpha_1,\dots, \alpha_k\}$$ denote by $$\pi_{\alpha, \beta}: I_\alpha \to I_\beta$$ the natural projection defined by $$\pi_{\alpha, \beta} (x_{\alpha_1},\dots,x_{\alpha_k})= (x_{\beta_1}, \dots, x_{\beta_s})$$.
For $$\omega$$ an analytic differential $$1$$-form in a neighbourhood $$U$$ of $$0\in \mathbb R^2$$ we have $$\Theta= \mathcal O_u\omega$$ the one-dimensional foliation generated by $$\omega$$ in $$U$$. If $$\gamma$$ is a leaf of $$\Theta$$, a Pfaffian arc ending at the origin, one may assume $$\gamma =\{ (x,\phi (x))\in\mathbb R^2\}$$, $$\phi :I_+ \to I$$ an analytic function such that $$\lim_{x \to 1}\phi =1$$, and $$\lim_{x \to 0}\frac{d\phi} {dx}=0$$. A $$C^1$$ extension of $$\phi$$ is associated to the couple $$(\omega, \gamma)$$, namely $$f:I\to I, f(t)=\phi(| t| )$$. For several couples $$(\omega_i, \gamma_i)$$, $$i=1,\dots,q$$ and the corresponding associates $$h_i:I \to I$$ we put $$h=(h_1,\dots,h_q):I \to I^q$$. In this situation, for any integer $$n\in \mathbb N$$, and for any increasing multi-index $$\alpha =(\alpha_1,\dots,\alpha_k)\in N_*^k$$, we define $$F_{(1,\dots,n)}: I^n \to I^{n+q}$$ by $$F_{(1,\dots,n)} (x_1,\dots,x_n)= (x_1,h(x_1), x_2,\dots,x_n)$$ and denote by $$F_\alpha$$ the restriction $$F_{(1,\dots,\alpha_k)}| I_\alpha$$.
We call basic sets the sets of the form $$\mathcal B (I_\alpha)=F_\alpha^{-1}\{E\cap F_\alpha(I_\alpha): E$$ subanalytic subset of some $$\mathbb R^N \supset F_\alpha (I_\alpha) \}$$. By definition $$A\subset I_\alpha$$ is $$T$$-constructible in $$I_\alpha$$ if $$A=\pi_{\alpha, \beta}(B)$$, for some $$B\in \mathcal B (I_\alpha)$$. A set $$X\subset I^n$$ is called a $$\mathcal D_n$$-set if there exists an increasing multi-index $$\alpha =(\alpha_1,\dots,\alpha_n)\in N_*^n$$ such that $$X=i_\alpha^{-1}(B)$$, $$B\subset I_\alpha$$, $$T$$-constructible and $$i_\alpha:I^n \to I_\alpha$$ is given by $$i_\alpha(e_j)= e_{\alpha_j}$$, $$j=1,\dots,n$$.
The main result of this paper asserts that the collection $$\{\mathcal D_n\}_{n\in \mathbb N_*}$$ is an $$O$$-minimal structure on the unit interval $$I$$. As a corollary the author proves that the complementary of a sub-Pfaffian set is also sub-Pfaffian.
##### MSC:
 32B20 Semi-analytic sets, subanalytic sets, and generalizations 53C12 Foliations (differential geometric aspects) 34C30 Manifolds of solutions of ODE (MSC2000)
##### Keywords:
Pfaffian systems; stratifications; subanalytic sets