The Aojasiewicz-Siciak Condition of the Pluricomplex Green Function

The aim of this paper is to address a problem raised originally by L. Gendre, later by W. PleA > niak and recently by L. Biaas-CieA1/4 and M. Kosek. This problem concerns the pluricomplex Green function and consists in finding new examples of sets with so-called Aojasiewicz-Siciak ((AS) for short) property. So far, the known examples of such sets are rather of particular nature. We prove that each compact subset of a"e (N) , treated as a subset of a", (N) , satisfies the Aojasiewicz-Siciak condition. We also give a sufficient geometric criterion for a semialgebraic set in a"e(2), but treated as a subset of a",, to satisfy this condition. This criterion applies more generally to a set in a", definable in a polynomially bounded o-minimal structure.