# Buchberger-Zacharias Theory of Multivariate Ore Extensions

## TL;DR

This paper extends Buchberger's theory and algorithms for Gr"obner bases to multivariate Ore extensions over rings that are modules over principal ideal domains, utilizing M"oller Lifting Theorem.

## Contribution

It introduces a Buchberger theory and algorithms specifically designed for multivariate Ore extensions, expanding the applicability of Gr"obner basis methods.

## Key findings

- Developed Buchberger theory for multivariate Ore extensions.
- Implemented algorithms based on M"oller Lifting Theorem.
- Enhanced computational methods for non-commutative polynomial rings.

## Abstract

We present Buchberger Theory and Algorithm of Gr\"obner bases for multivariate Ore extensions of rings presented as modules over a principal ideal domain. The algorithms are based on M\"oller Lifting Theorem.