Monomial methods in iterated local skew power series rings

TL;DR

This paper develops monomial orderings and Gröbner basis theory for iterated local skew power series rings over finite fields and p-adic integers, establishing properties like polynormality and unique factorization domains.

Contribution

It introduces the existence of monomial orders and Gröbner bases in these rings, and proves properties such as polynormality and unique factorization in specific cases.

Findings

All rank-2 local skew power series rings over _p are polynormal.

A rank-2 local skew power series ring over _p can be a UFD in the sense of Chatters-Jordan.

Existence of monomial orders under mild conditions.

Abstract

Let $A = \mathbb{F}_p$ or $\mathbb{Z}_p$, and let $R = A[[x_1]][[x_2; \sigma_2, \delta_2]]\dots[[x_n;\sigma_n,\delta_n]]$, an iterated local skew power series ring over $A$. Under mild conditions, we show that (multiplicative) monomial orders exist, and develop the theory of Gr\"obner bases for $R$. We show that all rank-2 local skew power series rings over $\mathbb{F}_p$ satisfy polynormality, and give an example of a rank-2 local skew power series ring over $\mathbb{Z}_p$ which is a unique factorisation domain in the sense of Chatters-Jordan.