Grothendieck rings of \(\mathbb{Z}\)-valued fields (Q2736587)

A \(\mathbb Z\)-valued field \(M\) is field with a valuation into an ordered group elementarily equivalent to the integers. The authors prove the existence of a definable bijection between \(M^2\) and \(M^2\setminus \{(0,0)\}\), under mild conditions on the basic language, which include the basic language for the classical local fields. This implies the triviality of the Grothendieck ring of the corresponding structure, which is analogous to the Grothendieck ring in algebraic K-theory, and has similar basic properties, and answer a question posed by J. Denef. In the case of the \(p\)-adic numbers, they show the existence of a definable bijection between the p-adic integers and the p-adic integers with one point removed, answering a question posed by the reviewer. The first author has proved the remarkable fact that in the \(p\)-adic numbers, the existence of a definable bijection between two definable sets is equivalent to the equality of their dimension, but this will appear elsewhere.