On Lojasiewicz's inequality and the Nullstellensatz for rings of semialgebraic functions

Abstract

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