Efficient Gr\"obner Bases Computation over Principal Ideal Rings

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 $R/nR$.