| Abstrakt: |
When k is a finite field, [J. Becker, J. Denef and L. Lipshitz, Further remarks on the elementary theory of formal power series rings, in Model Theory of Algebra and Arithmetic, Proceedings Karpacz, Poland, Lecture Notes in Mathematics, Vol. 834 (Springer, Berlin, 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 restricts 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 (k ((t)) , res) is undecidable whenever k is an infinite field. As a consequence, we get that (ℂ ((t)) , Res 0) is undecidable, where Res 0 : f ↦ Res 0 (f) maps f to its complex residue at 0. [ABSTRACT FROM AUTHOR] |
