Homogeneous minimal polynomials with prescribed interpolation conditions

Abstract:

Given a compact set $E$ in $\mathbb {C}^{N+1}$, we consider the problem of finding a homogeneous polynomial of degree $d$ on $\mathbb {C}^{N+1}$ which deviates the least from zero on $E$ with respect to the uniform norm among all those satisfying interpolation conditions of the form $p(a)=f(a)$ where $a$ belongs to a given finite subset of $\mathbb {C}^{N+1}$ and $f$ is any function on such set. We show that this formalism enables one to recover several types of minimal polynomials previously studied and to prove a general invariance property of such minimal polynomials under polynomial mappings.