Widom factors in $\mathbb C^n$

Abstract

We generalize the theory of Widom factors to the $\mathbb C^n$ setting. We define Widom factors of compact subsets $K\subset \mathbb C^n$ associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets $K=K_1\times\cdots\times K_n$ of $\mathbb C^n$, where each $K_j$ is a non-polar compact subset of $\mathbb C$, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each $K_j$ is a subset of the real line, we provide improved lower bounds for Widom factors for some weight functions $w$; in particular, for the case $w\equiv 1$. Finally, we define the Mahler measure of a multivariate polynomial relative to $K\subset \mathbb C^n$ and obtain lower bounds for this quantity on product sets.