Advances and improvements in the theory of standard bases and syzygies

DOI10.1007/BF01273348zbMath0854.13015OpenAlexW2056117826MaRDI QIDQ1924596

Gert-Martin Greuel, Gerhard Pfister

Publication date: 22 November 1996

Published in: Archiv der Mathematik (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/bf01273348

zbMATH Keywords

multiplicityGröbner basisdimensiondeformationHilbert functionstandard basistangent cone algorithmideal in a polynomial ringcomputer algebra system SINGULARHilbert's syzygy theoreminvariants of the local ring of an algebraic variety

Mathematics Subject Classification ID

Singularities in algebraic geometry (14B05) Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) (13P10) Computational aspects in algebraic geometry (14Q99)

Related Items

Topological Equisingularity: Old Problems from a New Perspective (with an Appendix by G.-M. Greuel and G. Pfister on SINGULAR), Standard bases in mixed power series and polynomial rings over rings, A normal form algorithm for the Brieskorn lattice, On an implementation of standard bases and syzygies in SINGULAR, The \(\lambda \)-Gröbner bases under polynomial composition, Tangent cone algorithm for homogenized differential operators, Toward a theory of monomial preorders, Newton's method with deflation for isolated singularities of polynomial systems, Standard bases in \(Kt_1,\ldots ,t_m[x_1,\ldots ,x_n^s\)], Reduction Mod p of Standard Bases, Algorithm computing the local \(b\) function by an approximate division algorithm in \(\widehat{\mathcal D}\), Evaluation techniques for zero-dimensional primary decomposition, A combinatorial approach to involution and \(\delta \)-regularity. I: Involutive bases in polynomial algebras of solvable type, Local equations for the toric Hilbert scheme, Computer algebra and algebraic geometry -- achievements and perspectives, Deformations with constant Lê numbers and multiplicity of nonisolated hypersurface singularities, Zariski's multiplicity question and aligned singularities

Uses Software

• CoCoA
• Macaulay2
• SINGULAR

Cites Work

• Constant Milnor number implies constant multiplicity for quasihomogeneous singularities
• New constructive methods in classical ideal theory
• La queste del Saint \(\text{Gr}_ a(\text{AL})\): A computational approach to local algebra
• The tangent cone algorithm and homogenization
• On an implementation of standard bases and syzygies in SINGULAR
• Topologically Trivial Deformations of Isolated Quasihomogeneous Hypersurface Singularities are Equimultiple
• Some open questions in the theory of singularities
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