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Abstract

We give a state-of-the-art survey of investigations concerning multivariate polynomial inequalities. A satisfactory theory of such inequalities has been developed due to applications of both the Gabrielov-Hironaka-Łojasiewicz subanalytic geometry and pluripotential methods based on the complex Monge-Ampère operator. Such an approach permits one to study various inequalities for polynomials restricted not only to nice (nonpluripolar) compact subsets of $\Bbb R^n$ or $\Bbb C^n$ but also their versions for pieces of semialgebraic sets or other “small” subsets of $\Bbb R^n$ ($\Bbb C^n$).