Convexité polynomiale, polyhèdres polynomiaux spéciaux et applications

Abstract

We generalize in C n Hilbert Lemniscate Theorem. More precisely, any polynomially convex compact subset K in C n can be approximated externally by special polynomial polyhedra P defined by proper polynomial mappings from C n to C n with ‘almost’ all their zeros in P . In the particular case where K is balanced, we can choose the polynomial mapping ‘almost’ homogeneous with a zero at the origin of multiplicity ‘almost’ equal to the degree. A first consequence of this generalization is a precise version of Runge's theorem in C n . A second application is an uniform approximation in C n , of the pluricomplex Green function with pole at infinity for a L -regular compact set K, by maximal plurisubharmonic functions in L + with isolated logarithmic poles. To cite this article: S. Nivoche, C. R. Acad. Sci. Paris, Ser. I 342 (2006).