Value-sets of polynomials at \(p\)-adic integers (Q431256)

Let \({\mathbb Z}_p\) be the ring of \(p\)-adic integers. The objective of this paper is to study the classification of subsets of \({\mathbb Z}_p\) which are images \(f({\mathbb Z}_p^r)\) of polynomial functions \(f: {\mathbb Z}_p^r\to {\mathbb Z}_p\) with \(f\in {\mathbb Q}_p[X_1,\ldots,X_r]\) and with \(f\in{\mathbb Z}_p [X_1,\ldots,X_r]\). The authors restrict themselves to the case of open sets and indeed compact--open since \({\mathbb Z}_p\) is compact. Their results are: (1) any compact--open set \(A\) can be obtained as \(A=f({\mathbb Z}_p^r)\) for some large enough \(r\) and \(f\in {\mathbb Z}_p[X_1, \ldots,X_r]\) and that for any given \(r\) there exist \(A\) that are not of the form \(f({\mathbb Z}_p^r)\) for any \(f\in{\mathbb Z}_p [X_1,\ldots,X_r]\); (2) for any compact--open subset \(A\subseteq {\mathbb Z}_p\), there exists \(f\in {\mathbb Q}_p[X]\) such that \(A=f({\mathbb Z}_p)\). In the last section, the authors present a necessary condition on \(A\) to be represented as \(A=f({\mathbb Z}_p)\) for some \(f\in {\mathbb Z}_p[X]\) by means of the usual \(p\)--adic Haar measure on \({\mathbb Z}_p\).