What are the necessary and sufficient conditions for an exponential function over the positive integers to always produce an integer?

Question

Let $f(n)=ab^{P(n)}$ be an exponential function with domain $Z^+$, real base $b$, real coefficient $a$ and $P(n)$ an arbitrary polynomial function of $n$. Then what are the necessary and sufficient conditions for the following to hold?

$f:Z^+\rightarrow Z$