Standard bases over Euclidean domains. (January 2021)

• Record Type: Journal Article
• Authors: Eder, Christian; Pfister, Gerhard; Popescu, Adrian
• Is Part Of: Journal of symbolic computation. Volume 102(2020)
• Page Start: 21
• Page End: 36
• Publication Date: 2021-01
• Subjects: Gröbner bases -- Standard bases -- Euclidean domains -- Algorithms; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2019.10.007
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: In this paper we state and explain techniques useful for the computation of strong Gröbner and standard bases over Euclidean domains: First we investigate several strategies for creating the pair set using an idea by Lichtblau. Then we explain methods for avoiding coefficient growth using syzygies. We give an in-depth discussion on normal form computation resp. a generalized reduction process with many optimizations to further avoid large coefficients. These are combined with methods to reach GCD-polynomials at an earlier stage of the computation. Based on various examples we show that our new implementation in the computer algebra system Singular is, in general, more efficient than other known implementations.