The Largest Possible Finite Degree of Functions between Commutative Groups

Abstract

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 largest 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.