On Łojasiewicz's inequality and the nullstellensatz for rings of semialgebraic functions (Q397873)

Let \(M\subset \mathbb{R}^n\) be a semialgebraic set and let \(f,g:M\to \mathbb{R}\) be continuous semialgebraic functions.NEWLINENEWLINEThe following classical Łojasiewicz inequality is well known: if \(M\) is locally compact and \(f^{-1}(0)\subset g^{-1}(0)\) then there exist a positive integer \(l\) and a continuous semialgebraic function \(h:M\to\mathbb{R}\) such that \(g^l=fh\). In particular \(|g(x)|^l\leq c|f(x)|\) for \(x\in M\), provided \(c=\sup\{|h(x)|:x\in M\}\) exists.NEWLINENEWLINEIn the paper, the authors generalize this result to arbitrary semialgebraic sets \(M\) (not necessarily locally compact) but only in the ring \(S^*(M)\) of continuous bounded semialgebraic functions on the set \(M\). The main result of the paper is Theorem~~1.3 (with equivalent formulation in Theorem~~3.10), which says: if \(f,g\in S^*(M)\) are functions such that each maximal ideal of \(S^*(M)\) containing \(f\) contains \(g\), too, then \(g^l=hf\) for suitable positive integer \(l\) and a function \(h\in S^*(M)\). In particular \(|g(x)|^l\leq (\sup_M(|h|))|f(x)|\) for \(x\in M\).NEWLINENEWLINEThe above Łojasiewicz inequality is used as the crucial tool to prove the Nullstellensatz for bounded and cotinuous semialgebraic functions (Corollary 3.9). The authors also prove the classical Łojasiewicz inequality. Moreover they show that without the assumption of local compactness of the set \(M\) the classical Łojasiewicz inequality in general fails.