Multivariate Polynomial Inequalities via Pluripotential Theory and Subanalytic Geometry Methods

In this paper, 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-Ã Lojasiewicz 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 Rn or Cn but also their versions for pieces of semialgebraic sets or other ”small ” subsets of Rn (Cn). I. Global polynomial inequalities Multivariate polynomial inequalities are closely related to the Siciak extremal function associated with a compact subset E of C, ΦE(z) = sup{|p(z)|1/ deg p}, where p : C → C is a nonconstant polynomial with sup |p|(E) ≤ 1, z ∈ C. Siciak’s function establishes an important link between polynomial approximation in several variables and pluripotential theory. This yields its numerous applications in complex and real analysis. It is known (Zakharyuta 1976, Siciak 1981) that log ΦE(z) = VE(z) := sup{u(z) : u ∈ L(Cn), u ≤ 0 on E}, where L(Cn) = {u ∈ PSH(C) : u(z) − log |z| ≤ O(1) as |z| → ∞} is the Lelong class of plurisubharmonic functions with logarithmic growth