Inner functions and zero sets for $\ell^{p}_{A}$

In this paper we characterize the zero sets of functions from $\ell^{p}_{A}$ (the analytic functions on the open unit disk $D$ whose Taylor coefficients form an $\ell^p$ sequence) by developing a concept of an "inner function" modeled by Beurling's discussion of the Hilbert space $\ell^{2}_{A}$ (the classical Hardy space). The zero set criterion is used to construct families of zero sets which are not covered by classical results. In particular, it is proved that when $p > 2$, there are zero sets for $\ell^{p}_{A}$ which are not Blaschke sequences.