Submission date: 4 March 2022

Abstract:

When k is a finite field, Becker-Denef-Lipschitz (1979) observed that the total residue map res: k((t)) → k, which picks out the constant term of the Laurent series, is definable in the language of rings with a parameter for t. Driven by this observation, we study the theory VF_{res,i} of valued fields equipped with a linear form res: K → k which specializes to the residue map on the valuation ring. We prove that VF_{res,i} does not admit a model companion. In addition, we show that the power series field (k((t)), res), equipped with such a total residue map, is undecidable whenever k is an infinite field. As a consequence, we get that (ℂ((t)), Res_0) is undecidable, where Res_0:ℂ((t)) → ℂ, f ↦ Res_0(f), maps f to its complex residue at 0.

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Full text arXiv 2203.02374: pdf, ps.