Sub-Pfaffian sets and a generalization of Wilkie's theorem (Q2739302)

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 \(\pi_{\alpha, \beta} (x_{\alpha_1},\dots,x_{\alpha_k})= (x_{\beta_1}, \dots, x_{\beta_s})\).NEWLINENEWLINE 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 \(F_{(1,\dots,\alpha_k)}| I_\alpha\).NEWLINENEWLINE 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 \(i_\alpha(e_j)= e_{\alpha_j}\), \(j=1,\dots,n\).NEWLINENEWLINEThe 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.