Approximation theorems for spaces of localities
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Mathematische Zeitschrift
Approximation theorems for spaces of localities Sylvy Anscombe1 · Philip Dittmann2 · Arno Fehm2 Received: 8 July 2019 / Accepted: 14 January 2020 © The Author(s) 2020
Abstract The classical Artin–Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence.
1 Introduction We fix a field K , elements x1 , . . . , xn ∈ K and z 1 , . . . , z n ∈ K × and start by recalling the classical approximation theorem for absolute values: Theorem 1.1 (Artin–Whaples 19451 ) Let |.|1 , . . . , |.|n be nontrivial absolute values on K . (I) Assume that |.|1 , . . . , |.|n are pairwise inequivalent. Then there exists x ∈ K with |x − xi |i < |z i |i for i = 1, . . . , n. Since the non-archimedean absolute values correspond to Krull valuations of rank 1, the following theorem is a generalization in the non-archimedean case: 1 Published in [3]. See also the historical remarks in [33, §4.2.1]. See also [22, XII.1.2], [10, 1.1.3].
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Philip Dittmann [email protected] Sylvy Anscombe [email protected] Arno Fehm [email protected]
1
Jeremiah Horrocks Institute, University of Central Lancashire, Preston PR1 2HE, UK
2
Institut für Algebra, Fakultät Mathematik, Technische Universität Dresden, 01062 Dresden, Germany
123
S. Anscombe et al.
Theorem 1.2 (Bourbaki2 ) Let v1 , . . . , vn be nontrivial valuations3 on K . (I) Assume that v1 , . . . , vn are pairwise independent.4 Then there exists x ∈ K with vi (x − xi ) > vi (z i ) for i = 1, . . . , n. In the literature, one can find three possible directions of generalizing Theorem 1.2. Firstly, one can unify Theorems 1.1 and 1.2 to approximate with respect to finitely many pairwise independent valuations and absolute values, like in [29, (4.2) Corollary] or [34, Cor. 27.14]. This also includes approximation with respect to (not necessarily archimedean) orderings on K , see Theorem 2.8 below. Secondly, one can relax the condition of pairwise independence in Theorem 1.2, at the expense of introducing a compatibility condition, as done by Nagata [24] and Ribenboim [30]. Theorem 1.3 (Ribenboim 19575 ) Let v1 , . . . , vn be valuations on K . (I) Assume that v1 , . . . , vn are pairwise incomparable and that if w is a common coarsening of vi and v j , i = j, then w(xi − x j ) ≥ w(z i ) = w
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