  We introduce and study the algebraic, analytic and lattice properties of
regular homogeneous polynomials and holomorphic functions on complex Banach
lattices. We show that the theory of power series with regular terms is closer
to the theory of functions of several complex variables than the theory of
holomorphic functions on Banach spaces. We extend the concept of the Bohr
radius to Banach lattices and show that it provides us with a lower bound for
the ratio between the radius of regular convergence and the radius of
convergence of a regular holomorphic function. This allows us to show that the
radius of regular convergence coincides with the radius of convergence for
holomorphic functions on finite dimensional spaces and orthogonally additive
holomorphic functions but that these radii can be radically different on
$\ell_p$ for $p>1$.

Functional Analysis