Valued fields with a total residue map

2022/03/04 by Kartas, Konstantinos
#FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2203.02374

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,ι 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,ι 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) ), Res₀) is undecidable, where Res₀:ℂ( (t) )→ ℂ:f↦ Res₀(f) maps f to its complex residue at 0.

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