Classification of semi-algebraic \(p\)-adic sets up to semi-algebraic bijection (Q2754271)

Let~\(K\) be a finite extension of~\({\mathbb Q}_p\). A subset of~\(K^n\) is said to be semi-algebraic if it is a finite Boolean combination of projections of affine \(K\)-varieties. A semi-algebraic subset of some~\(K^n\) has a well defined dimension [see \textit{P. Scowcroft} and \textit{L. van den Dries}, J. Symb. Logic 53, No. 4, 1138-1164 (1988; Zbl 0692.14014)]. NEWLINENEWLINENEWLINEIn the paper under review, the author proves that two infinite semi-algebraic subsets are semi-algebraically isomorphic if and only if they have the same dimension. The main ingredient is a ''rectilinearization'' theorem.