Valued fields with a total residue map

When [Formula: see text] 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 Algebra Arithmetic, Proceedings Karpacz, Poland, Lecture Notes Mathematics, Vol. 834 (Springer, Berlin, 1979)] observed that total residue map text], which picks out constant term Laurent series, definable language rings with parameter for text]. Driven by this observation, we study valued fields equipped linear form restricts to valuation ring. We prove does not admit model companion. In addition, show undecidable whenever an infinite field. As consequence, get undecidable, where maps its complex at