An exponential Lojasiewicz inequality for semi-Pfaffian sets

Abstract

We prove a generalization of the version, due to A. Gabrielov, of the {\L}ojasiewicz inequality for not necessarily restricted semi-Pfaffian sets. Unlike the most variants of the {\L}ojasiewicz inequality, it describes the rate of growth of a Pfaffian function on a semi-Pfaffian set $X$ not only in a neighbourhood of its zero set in the domain of $X$ but also in a neighbourhood of zeroes on the boundary of the domain. Gabrielov's version is essentially one-dimensional, we generalize it to a multidimensional case.