Affiliation:
1. School of Mathematics , Shandong University , Jinan 250100 , P. R. China
Abstract
Abstract
This paper presents a modified version of Leibman’s group-theoretic generalizations of the difference calculus for polynomial maps from nonempty commutative semigroups to groups, and proves that it has many desirable formal properties when the target group is locally nilpotent and also when the semigroup is the set of nonnegative integers.
We will apply it to solve Waring’s problem for general discrete Heisenberg groups in a sequel to this paper.
Funder
National Science Foundation
Subject
Algebra and Number Theory
Reference19 articles.
1. E. Aichinger and J. Moosbauer,
Chevalley–Warning type results on abelian groups,
J. Algebra 569 (2021), 30–66.
2. P.-J. Cahen and J.-L. Chabert,
Integer-Valued Polynomials,
Math. Surveys Monogr. 48,
American Mathematical Society, Providence, 1997.
3. P. L. Clark and U. Schauz,
Functional degrees and arithmetic applications I: The set of functional degrees,
J. Algebra 608 (2022), 691–718.
4. P. L. Clark and U. Schauz,
Functional degrees and arithmetic applications II: The group-theoretic prime Ax–Katz theorem, preprint (2023), https://arxiv.org/abs/2305.01304.
5. A. E. Clement, S. Majewicz and M. Zyman,
The Theory of Nilpotent Groups,
Birkhäuser/Springer, Cham, 2017.