  We consider maps between commutative groups and their functional degrees.
These degrees are defined based on a simple idea -- the functional degree
should decrease if a discrete derivative is taken. We show that the maps of
finite functional degree are precisely the maps that can be written as
polyfracts, as polynomials in several variables but with binomial functions in
the place of powers. Moreover, the degree of a polyfract coincides with its
functional degree. We use this to determine the biggest possible finite
functional degree that the maps between two given finite commutative groups can
have. This also yields a solution to Aichinger and Moosbauer's problem of
finding the nilpotency degree of the augmentation ideal of the group ring
$Z_{p^\beta}[Z_{p^{\alpha_1}}\times Z_{p^{\alpha_2}}\times\dots\times
Z_{p^{\alpha_n}}]$. Some generalizations and simplifications of proofs to
underlying facts are presented, too.
