Polynomial inequalities, o-minimality and Denjoy–Carleman classes

Abstract

We study several intimately related problems in the theory of multivariate polynomial inequalities. Firstly, given a map h in certain quasianalytic Denjoy–Carleman classes, we show how to decide whether the image under h of a set satisfying Markov's (resp. Nikolskii's) inequality satisfies Markov's (resp. Nikolskii's) inequality. Our approach relies heavily on the theory of o-minimal structures, particularly on the work of Rolin, Speissegger and Wilkie. Secondly, we establish the relation between Markov's inequality and Nikolskii's inequality (both in general setting and in o-minimal setting). Thirdly, we prove that each compact, definable (in an o-minimal structure) set satisfying Markov's inequality is fat. In particular, this solves in the o-minimal category the well-known problem of nonpluripolarity of sets satisfying Markov's inequality. And lastly, we develop a unified method of polynomial approximation for various classes of smooth functions of several variables.