Widom factors in C<i><SUP>n</SUP></i>

We generalize the theory of Widom factors to the C-n setting. We define Widom factors of compact subsets K subset of C-n associated with multivariate orthogonal polynomials and weighted Chebyshev polynomials. We show that on product subsets K = K-1 x ... x K-n of C-n, where each K-j 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 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 equivalent to 1. Finally, we define the Mahler measure of a multivariate polynomial relative to K subset of C-n and obtain lower bounds for this quantity on product sets. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.