TY - JOUR

T1 - Functional degrees and arithmetic applications I

T2 - The set of functional degrees

AU - Schauz, Uwe

AU - Clark, Pete L.

N1 - Publisher Copyright:
© 2022 Elsevier Inc.

PY - 2022/10/15

Y1 - 2022/10/15

N2 - We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any finitely generated commutative groups A and B, we compute the complete set D(A,B) of functional degrees of all maps between A and B. In particular, we see how big the functional degree of a function with finite functional degree can get, in which cases there is a maximal finite functional degree, and when there are functions of infinite functional degree between A and B. This yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form Zpβ[Zpα1×Zpα2×⋯×Zpαn] with p,β,n,α1,…,αn∈Z+, p prime.

AB - We give a further development of the Aichinger-Moosbauer calculus of functional degrees of maps between commutative groups. For any finitely generated commutative groups A and B, we compute the complete set D(A,B) of functional degrees of all maps between A and B. In particular, we see how big the functional degree of a function with finite functional degree can get, in which cases there is a maximal finite functional degree, and when there are functions of infinite functional degree between A and B. This yields a solution to Aichinger and Moosbauer's problem of finding the nilpotency index of the augmentation ideal of group rings of the form Zpβ[Zpα1×Zpα2×⋯×Zpαn] with p,β,n,α1,…,αn∈Z+, p prime.

KW - Abelian groups

KW - Functional degree

KW - Polynomial functions

UR - http://www.scopus.com/inward/record.url?scp=85134700602&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2022.05.035

DO - 10.1016/j.jalgebra.2022.05.035

M3 - Article

AN - SCOPUS:85134700602

SN - 0021-8693

VL - 608

SP - 691

EP - 718

JO - Journal of Algebra

JF - Journal of Algebra

ER -