Characterizing Diophantine Henselian valuation rings and valuation ideals

Anscombe, Sylvy orcid iconORCID: 0000-0002-9930-2804 and Fehm, Arno (2017) Characterizing Diophantine Henselian valuation rings and valuation ideals. Proceedings of the London Mathematical Society, 115 (2). pp. 293-322. ISSN 0024-6115

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We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal.

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