Non-existence of the Artin function for Henselian pairs (Q1337531)

If \((A,m)\) is a local Henselian ring, the strong version of the Artin approximation theorem says that given a system of algebraic equations over \(A\), a sufficiently good approximate solution in the \(m\)-adic topology can be approximated by an exact solution. The Artin function is a quantitative measure of ``sufficiently good'' in the above statement. The main point of this paper is to give a simple counterexample to the analogue of the above statement for Henselian pairs \((A,I)\).