Unconditionality for m-homogeneous polynomials on l(infinity)n

Abstract

[EN] Let x(m, n) be the unconditional basis constant of the monomial basis Z alpha, alpha is an element of N(0)n with vertical bar alpha vertical bar = m, of the Banach space of all m-homogeneous polynomials inn complex variables, endowed with the supremum norm on the n-dimensional unit polydisc Dn. We prove that the quotient of sup(m) m root sup(m)chi(m, n) and root n/log n tends to 1 as n -> infinity. This reflects a quite precise dependence of chi(m, n) on the degree m of the polynomials and their number n of variables. Moreover, we give an analogous formula for m-linear forms, a reformulation of our results in terms of tensor products, and as an application a solution for a problem on Bohr radii.