Let χ(m,n) be the unconditional basis constant of the monomial basis $z^{\alpha }$, α ∈ ℕ₀ⁿ with |α| = m, of the Banach space of all m-homogeneous polynomials in n complex variables, endowed with the supremum norm on the n-dimensional unit polydisc ⁿ. We prove that the quotient of $su{p}_m\sqrt[m]{su{p}_m\chi (m,n)}$ and √(n/log n) tends to 1 as n → ∞. This reflects a quite precise dependence of χ(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...

Our aim is to introduce a new notion of unconditionallity, in the context of polynomials in Banach spaces, that looks directly to the polynomial topology defined on the involved spaces. This notion allows us to generalize some well-known relations of duality that appear in the linear context.