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
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[1] Berkesch, C.; Schreyer, F.-O., Syzygies, finite length modules, and random curves, (2014)
[2] Chiodo, A.; Eisenbud, D.; Farkas, G.; Schreyer, F.-O., Syzygies of torsion bundles and the geometry of the level modular variety over \(\overline{\mathcal{M}}_g\), Invent. Math., 194, 1, 73-118, (2013) · Zbl 1284.14006
[3] Decker, W.; Greuel, G.-M.; Pfister, G.; Schönemann, H., {\scsingular} 4-0-2 - a computer algebra system for polynomial computations, (2014)
[4] Dumas, J.-G.; Giorgi, P.; Pernet, C., Dense linear algebra over word-size prime fields: the FFLAS and FFPACK packages, ACM Trans. Math. Softw., 35, 3, 1-42, (2008)
[5] Eisenbud, D., Commutative algebra with a view toward algebraic geometry, (1995), Springer · Zbl 0819.13001
[6] Grayson, D.; Stillman, M., Macaulay2, a software system for research in algebraic geometry (version 1.7), (2014)
[7] Greuel, G.-M.; Pfister, G., A {\scsingular} introduction to commutative algebra, (2007), Springer
[8] La Scala, R.; Stillman, M., Strategies for computing minimal free resolutions, J. Symb. Comput., 26, 4, 409-431, (1998) · Zbl 1034.68716
[9] Motsak, O., . A {\scsingular} 4-0-2 library for computing the schreyer resolution of ideals and modules, (2014)
[10] Oeding, L.; Ottaviani, G., Eigenvectors of tensors and algorithms for Waring decomposition, J. Symb. Comput., 54, 9-35, (2013) · Zbl 1277.15019
[11] Ranestad, K.; Schreyer, F.-O., Varieties of sums of powers, J. Reine Angew. Math., 525, 147-181, (2000) · Zbl 1078.14506
[12] Schreyer, F.-O., Die berechnung von syzygien mit dem verallgemeinerten weierstrasschen divisionssatz, (1980), Diplomarbeit Hamburg
[13] Schreyer, F.-O., A standard basis approach to syzygies of canonical curves, J. Reine Angew. Math., 421, 83-123, (1991) · Zbl 0729.14021
[14] Stein, W. A., Sage mathematics software (version 6.4.1). the sage development team, (2014)
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