Definability and decidability in global and local fields
Final Report Abstract
A mathematical problem is called decidable if, roughly speaking, there exists an algorithm that computes the correct answer for each input. Questions on decidability in number theory have a long history going back at least to Hilbert’s Tenth Problem on solvability of diophantine equations: Does there exist an algorithm that computes whether a given integer polynomial has an integer zero? Nowadays this is a lively and highly interdisciplinary research area with influences from number theory, arithmetic geometry, Galois theory, model theory (mathematical logic) and computational complexity theory (computer science). Important open questions here are the decidability of the existential theory of the rational numbers (i.e., does there exist an algorithm that decides whether a given polynomial has a rational zero) and other so-called global fields, and the decidability of the full first-order theory of local fields in positive characteristic, i.e. power series fields over finite fields. Very often, decidability problems in this area are closely linked to questions of definability: Which subsets of a ring or field are diophantine, that is, projection of the zero set of a polynomial, or, more generally, definable by a first-order formula in the language of rings? This project contributed to questions of definability and decidability in global and local fields, more generally in algebraic fields, function fields, and henselian valued field. Some of the highlights include results on existential definability in power series fields over algebraic fields, insights into the existential theory of global fields, a topological description of definable sets in rational function fields over local fields, a first-order characterization of each ring finitely generated over a local field (within that class), and the development of a model theory of henselian valued fields in positive characteristic assuming resolution of singularities.
Publications
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Approximation theorems for spaces of localities. Mathematische Zeitschrift, 296(3-4), 1471-1499.
Anscombe, Sylvy; Dittmann, Philip & Fehm, Arno
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Ap‐adic analogue of Siegel's theorem on sums of squares. Mathematische Nachrichten, 293(8), 1434-1451.
Anscombe, Sylvy; Dittmann, Philip & Fehm, Arno
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Denseness results in the theory of algebraic fields. Annals of Pure and Applied Logic, 172(8), 102973.
Anscombe, Sylvy; Dittmann, Philip & Fehm, Arno
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Existential rank and essential dimension of diophantine sets.
Daans, N.; Dittmann, Philip & Fehm, Arno
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Nondefinability of Rings of Integers in Most Algebraic Fields. Notre Dame Journal of Formal Logic, 62(3).
Dittmann, Philip & Fehm, Arno
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On the Northcott property and local degrees. Proceedings of the American Mathematical Society, 149(6), 2403-2414.
Checcoli, S. & Fehm, A.
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Axiomatizing the existential theory of q((t)). Algebra & Number Theory, 17(11), 2013-2032.
Anscombe, Sylvy; Dittmann, Philip & Fehm, Arno
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Defect extensions and a characterization of tame fields. Journal of Algebra, 630, 68-91.
Rzepka, Anna & Szewczyk, Piotr
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Ax–Kochen–Ershov principles for finitely ramified henselian fields. Transactions of the American Mathematical Society.
Anscombe, Sylvy; Dittmann, Philip & Jahnke, Franziska
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Definable henselian valuations in positive residue characteristic. The Journal of Symbolic Logic, 1-26.
Ketelsen, Margarete; Ramello, Simone & Szewcyk, Poitr
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Interpretations of syntactic fragments of theories of fields.
Anscombe, Sylvy & Fehm, Arno
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On the existential theory of the completions of a global field.
Dittmann, Philip & Fehm, Arno
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Model theory of rings and fields finitely generated over local fields. PhD thesis, Dresden
Vollprecht, A.
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Universal-existential theories of fields. Proceedings of the Edinburgh Mathematical Society, 69(1), 95-124.
Anscombe, Sylvy & Fehm, Arno
