Abstract

Journal of Mathematical Logic, Volume 24, Issue 03, December 2024. 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 [math], 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 [math] of valued fields equipped with a linear form [math] which restricts to the residue map on the valuation ring. We prove that [math] does not admit a model companion. In addition, we show that [math] is undecidable whenever k is an infinite field. As a consequence, we get that [math] is undecidable, where [math] maps f to its complex residue at 0.