Functional Degrees And Arithmetic Applications, I: The Set Of Functional Degrees

Abstract

We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any fixed given commutative groups $A$ and $B$, we compute the largest possible finite functional degree that a map $f:A⟶B$ can have. We also determine the set of all possible degrees of such maps. This also yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form $Z_{p^{\beta }}[Z_{p^{{\alpha }_1}}\times Z_{p^{{\alpha }_2}}\times \cdots \times Z_{p^{{\alpha }_n}}]$ with $p,\beta ,n,{\alpha }_1,\dots ,{\alpha }_n\in {\mathbb{Z}}^{+}$, $p$ prime.

21 pages