Monomialization of morphisms and \(p\)-adic quantifier elimination (Q2839340)

The paper provides a new and short proof of Macintyre's Theorem on quantifier elimination for \(p\)-adic numbers. Macintyre's result can be reformulated in algebraic terms as follows: NEWLINENEWLINEIf \(f: X \to Y\) is a morphism of schemes of finite type over the field \(\mathbb Q_p\) of \(p\)-adic numbers (\(p\) a prime), then for every semi-algebraic subset \(A\) of \(X(\mathbb Q_p)\) the image \(f(A)\) is also a semi-algebraic subset of \(Y(\mathbb Q_p)\). NEWLINENEWLINEThis is the statement proved in this paper. To do that, the author uses a version of monomialization that follows directly from the Weak Toroidalization Theorem of \textit{D. Abramovich} and \textit{K. Karu} [Invent. Math. 139, No. 2, 241--273 (2000; Zbl 0958.14006)], actually from an extension of this result to non-closed fields obtained by the author himself just with \textit{D. Abramovich} and \textit{K. Karu} [Manuscr. Math. 142, No. 1-2, 257--271 (2013; Zbl 1279.14020)].