  Let $\mathcal F$ be either the set of all bounded holomorphic functions or
the set of all $m$-homogeneous polynomials on the unit ball of $\ell\_r$. We
give a systematic study of the sets of all $u\in\ell\_r$ for which the monomial
expansion $\sum\_{\alpha}\frac{\partial^\alpha f(0)}{\alpha !}u^\alpha$ of
every $f\in\mathcal F$ converges. Inspired by recent results from the general
theory of Dirichlet series, we establish as our main tool, independently
interesting, upper estimates for the unconditional basis constants of spaces of
polynomials on $\ell\_r$ spanned by finite sets of monomials.
