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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.
32B20 Semi-analytic sets, subanalytic sets, and generalizations
53C12 Foliations (differential geometric aspects)
34C30 Manifolds of solutions of ODE (MSC2000)