Ax-Schanuel and o-minimality (Q2825775)

Commonly known as Ax-Schanuel theorem, due to \textit{J. Ax} [Ann. of Math. (2) 93, 252--268 (1971; Zbl 0232.10026)], is the following result: If \(f_1,\ldots,f_n\) \(\in\mathbb{C}[[t_1,\ldots,t_m]]\) are power series that are \(\mathbb{Q}\)-linearly independent modulo \(\mathbb{C}\), then \(\dim_{\mathbb{C}}\mathbb{C}(f_1,\ldots,f_n,e(f_1),\ldots,e(f_n))\geq n+\mathrm{rank}(\partial f_i/\partial t_j)_{1\leq i\leq n,1\leq j\leq m}\) holds, where \(e(x):=e^{2\pi ix}\) and \(\dim_KL\) denotes the transcendence degree of \(L\) over \(K\).NEWLINENEWLINEThe main aim of this note is to give two equivalent geometric interpretations of this theorem and to present a model theoretical proof of one of these using the techniques of Pila-Zannier.NEWLINENEWLINEFor the entire collection see [Zbl 1326.11003].