Monomialization of morphisms and \(p\)-adic quantifier elimination. (English) Zbl 1274.14026
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:
If \(f: X \to Y\) 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 \(Y(\mathbb Q_p)\).
This 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 D. Abramovich and 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 D. Abramovich and K. Karu [Manuscr. Math. 142, No. 1-2, 257–271 (2013; Zbl 1279.14020)].
If \(f: X \to Y\) 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 \(Y(\mathbb Q_p)\).
This 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 D. Abramovich and 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 D. Abramovich and K. Karu [Manuscr. Math. 142, No. 1-2, 257–271 (2013; Zbl 1279.14020)].
Reviewer: Carlo Toffalori (Camerino)
MSC:
| 14G20 | Local ground fields in algebraic geometry |
| 11S05 | Polynomials |
| 11G25 | Varieties over finite and local fields |
| 11U09 | Model theory (number-theoretic aspects) |
| 03C10 | Quantifier elimination, model completeness, and related topics |
Keywords:
\(p\)-adic number; Quantifier elimination; Monomialization; Dominant morphism; Scheme of finite type; ToroidalizationReferences:
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