2019/06/20 by Eder, Christian, Hofmann, Tommy · 1 citation
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1906.08543
In this paper we present a new efficient variant to compute strong Gröbner 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 Gröbner basis computations to Gröbner 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 Gröbner 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 Gröbner basis is already a strong Gröbner basis for the input ideal over R/nR.