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Refined algorithms to compute syzygies. (English) Zbl 1405.14138
Summary: Based on the third author’s algorithm [Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrasschen Divisionssatz. Hamburg: Hamburg University. (Diploma Thesis) (1980); J. Reine Angew. Math. 421, 83–123 (1991; Zbl 0729.14021); C. Berkesch and the third author, Math. Sci. Res. Inst. Publ. 67, 25–52 (2015; Zbl 1359.13029)], we present two refined algorithms for the computation of syzygies. The two main ideas of the first algorithm, called LiftHybrid, are the following: first, we may leave out certain terms of module elements during the computation which do not contribute to the result. These terms are called “lower order terms”, see Definition 4.2. Second, we do not need to order the remaining terms of these module elements during the computation. This significantly reduces the number of monomial comparisons for the arithmetic operations. For the second algorithm, called LiftTree, we additionally cache some partial results and reuse them at the remaining steps.

14Q99 Computational aspects in algebraic geometry
13D02 Syzygies, resolutions, complexes and commutative rings
13P20 Computational homological algebra
Full Text: DOI arXiv
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