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Images of additive polynomials in \(\mathbb{F}_q((t))\) have the optimal approximation property. (English) Zbl 1009.12008

Let \(S\) be a non-empty subset of a valued field \((K,v).\) The set \(S\) is said to have the optimal approximation property (OA) in \((K,v)\) if for every point in \(K\) there exists a (not necessarily unique) closest point in \(S.\) The question arises whether the image of \(K\) under a polynomial \(f\in K[X_1, \dots, X_n]\) has OA. A polynomial \(f\) is called additive if \[ f(a_1+ b_1,\dots, a_n+b_n)= f(a_1, \dots, a_n)+f(b_1, \dots, b_n) \] for all elements \(a_1, \dots, a_n\), \(b_1, \dots, b_n\) in any extension field of \(K.\) A valued field is called maximal if it has no proper field extension preserving both value group and residue field. In [F. V. Kuhlmann, J. Symb. Logic 66, 771-791 (2001; Zbl 0992.03046)] it is proved that if \((K,v)\) is maximal then under a certain elementar condition on \(f\) the image of \(K\) has OA. Let \(\mathbb{F}_q\) denote the field with \(q\) elements, where \(q\) is a power of a prime \(p.\) The power series field \(\mathbb{F}_q ((t))\) carries a canonical valuation \(v_t\) with value group \(\mathbb{Z}\) and \(v_t(t) =1.\) The main result of the paper under review is Theorem 1: If \(f\) is an additive polynomial in several variables with coefficients in \(\mathbb{F}_q ((t))\), then the image of \(\mathbb{F}_q ((t))\) under \(f\) has the optimal approximation property in \((\mathbb{F}_q ((t)),v_t)\).
Reviewer: G.Pestov (Tomsk)

MSC:

12J25 Non-Archimedean valued fields
12L12 Model theory of fields
03C60 Model-theoretic algebra

Citations:

Zbl 0992.03046
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