On Łojasiewiczʼs inequality and the Nullstellensatz for rings of semialgebraic functions

In this article we present versions of Łojasiewiczʼs inequality and the Nullstellensatz for the ring of bounded semialgebraic functions on an arbitrary semialgebraic set M. We also prove that the classical Łojasiewicz inequality and Nullstellensatz for the ring of semialgebraic functions on a semialgebraic set M work if and only if M is locally compact.