  Some polynomials $P$ with rational coefficients give rise to well defined
maps between cyclic groups, $\Z_q\longrightarrow\Z_r$, $x+q\Z\longmapsto
P(x)+r\Z$. More generally, there are polynomials in several variables with
tuples of rational numbers as coefficients that induce maps between commutative
groups. We characterize the polynomials with this property, and classify all
maps between two given finite commutative groups that arise in this way. We
also provide interpolation formulas and a Taylor-type theorem for the
calculation of polynomials that describe given maps.
