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AN EXISTENTIAL ∅-DEFINITION OF $F_q [[t]]$ IN $F_q \left( t \right)$

  • WILL ANSCOMBE (a1) and JOCHEN KOENIGSMANN (a2)
Abstract

We show that the valuation ring $F_q [[t]]$ in the local field $F_q \left( t \right)$ is existentially definable in the language of rings with no parameters. The method is to use the definition of the henselian topology following the work of Prestel-Ziegler to give an ∃- $F_q $ -definable bounded neighbouhood of 0. Then we “tweak” this set by subtracting, taking roots, and applying Hensel’s Lemma in order to find an ∃- $F_q $ -definable subset of $F_q [[t]]$ which contains $tF_q [[t]]$ . Finally, we use the fact that $F_q $ is defined by the formula $x^q - x = 0$ to extend the definition to the whole of $F_q [[t]]$ and to rid the definition of parameters.

Several extensions of the theorem are obtained, notably an ∃-∅-definition of the valuation ring of a nontrivial valuation with divisible value group.

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[1]Ax, James, On the undecidability of power series fields. Proceedings of the American Mathematical Society, vol. 16 (1965), p. 846.
[2]Cluckers, Raf, Derakhshan, Jamshid, Leenknegt, Eva, and Macintyre, Angus, Uniformly defining valuation rings in henselian valued fields with finite and pseudo-finite residue field. Annals of Pure and Applied Logic, vol.164 (2013), pp. 12361246.
[3]Denef, Jan and Schoutens, Hans, On the decidability of the existential theory of . Fields Institute Communications, vol. 33 (2003), pp. 4360.
[4]Koenigsmann, Jochen, Elementary characterization of fields by their absolute Galois group. Siberian Advances in Mathematics, vol. 14 (2004), pp. 1642.
[5]Macintyre, Angus, On definable subsets of p-adic fields, this Journal, vol. 41 (1976), pp. 605610.
[6]Prestel, Alexander, Algebraic number fields elementarily determined by their absolute Galois group. Israel Journal of Mathematics, vol. 73 (1991), no. 2, pp. 199205.
[7]Prestel, Alexander and Ziegler, Martin, Model-theoretic methods in the theory of topological fields. Journal für die Reine und Angewandte Mathematik, vol. 299 (1978), no. 300, pp. 318341.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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