Widom factors in ℂn

Official URL: https://dx.doi.org/10.1016/j.jat.2025.106227

We generalize the theory of Widom factors to the ℂn setting. We define Widom factors of compact subsets K⊂ℂn associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets K=K1×⋯×Kn of ℂn, where each Kj is a non-polar compact subset of ℂ, these quantities have universal lower bounds which directly extend one dimensional results. Under the additional assumption that each Kj 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≡1. Finally, we define the Mahler measure of a multivariate polynomial relative to K⊂ℂn and obtain lower bounds for this quantity on product sets.