Franz Halter-Koch | Searcharxiv

arXiv2024

P-adic approximation of algebraic integers and residue class rings of rings of integer-valued polynomials

Let F:K be a Galois extension of number fields and Q a prime ideal of O_F lying over the prime P of O_K. By analyzing the Q-adic closure of O_K in O_F we characterize those rings of integers O_K for which every residue class ring of Int(O_K) modulo a non-zero prime ideal is GE2 (meaning that every unimodular pair can be trasformed to (1,0) by a series of elementary transformations).

math.NT↗

arXiv2022

On monoids of weighted zero-sum sequences and applications to norm monoids in Galois number fields and binary quadratic forms

Let $G$ be an additive finite abelian group and $Γ\subset \operatorname{End} (G)$ be a subset of the endomorphism group of $G$. A sequence $S = g_1 \cdot \ldots \cdot g_{\ell}$ over $G$ is a ($Γ$-)weighted zero-sum sequence if there are $γ_1, \ldots, γ_{\ell} \in Γ$ such that $γ_1 (g_1) + \ldots + γ_{\ell} (g_{\ell})=0$. We construct transfer homomorphisms from norm monoids (of Galois algebraic number fields with Galois group $Γ$) and from monoids of positive integers, represented by binary quadratic forms, to monoids of weighted zero-sum sequences. Then we study algebraic and arithmetic properties of monoids of weighted zero-sum sequences.

math.NT↗