# Efficient Gr\"obner Bases Computation over Principal Ideal Rings

## TL;DR

This paper introduces an efficient method for computing strong Gr"obner bases over quotients of principal ideal rings, leveraging a recursive reduction to computations over fields for squarefree moduli.

## Contribution

The authors present a novel lifting process that reduces Gr"obner basis computations over quotients of principal ideal rings to simpler computations over fields, improving efficiency.

## Key findings

- Reduces complex computations to simpler field-based calculations
- Allows recursive reduction for squarefree moduli
- Provides a practical approach for strong Gr"obner basis computation

## Abstract

In this paper we present a new efficient variant to compute strong Gr\"obner basis over quotients of principal ideal domains. We show an easy lifting process which allows us to reduce one computation over the quotient $R/nR$ to two computations over $R/aR$ and $R/bR$ where $n = ab$ with coprime $a, b$. Possibly using available factorization algorithms we may thus recursively reduce some strong Gr\"obner basis computations to Gr\"obner basis computations over fields for prime factors of $n$, at least for squarefree $n$. Considering now a computation over $R/nR$ we can run a standard Gr\"obner basis algorithm pretending $R/nR$ to be field. If we discover a non-invertible leading coefficient $c$, we use this information to try to split $n = ab$ with coprime $a, b$. If no such $c$ is discovered, the returned Gr\"obner basis is already a strong Gr\"obner basis for the input ideal over $R/nR$.