Book contents
- Frontmatter
- Contents
- Preface
- Invited Lectures
- List of Participants
- Aspects of Gert-Martin Greuel's Mathematical Work
- Exterior Algebra Methods for the Construction of Rational Surfaces in the Projective Fourspace
- Superisolated Surface Singularities
- Linear Free Divisors and Quiver Representations
- Derived Categories of Modules and Coherent Sheaves
- Monodromy
- Algorithmic Resolution of Singularities
- Newton Polyhedra of Discriminants: A Computation
- Depth and Differential Forms
- The Geometry of the Versal Deformation
- 21 Years of SINGULAR Experiments in Mathematics
- The Patchworking Construction in Tropical Enumerative Geometry
- Adjunction Conditions for One-Forms on Surfaces in Projective Three-Space
- Sextic Surfaces with Ten Triple Points
- Sextic Surfaces with 10 Triple Points
- Topology, Geometry, and Equations of Normal Surface Singularities
21 Years of SINGULAR Experiments in Mathematics
Published online by Cambridge University Press: 11 November 2009
- Frontmatter
- Contents
- Preface
- Invited Lectures
- List of Participants
- Aspects of Gert-Martin Greuel's Mathematical Work
- Exterior Algebra Methods for the Construction of Rational Surfaces in the Projective Fourspace
- Superisolated Surface Singularities
- Linear Free Divisors and Quiver Representations
- Derived Categories of Modules and Coherent Sheaves
- Monodromy
- Algorithmic Resolution of Singularities
- Newton Polyhedra of Discriminants: A Computation
- Depth and Differential Forms
- The Geometry of the Versal Deformation
- 21 Years of SINGULAR Experiments in Mathematics
- The Patchworking Construction in Tropical Enumerative Geometry
- Adjunction Conditions for One-Forms on Surfaces in Projective Three-Space
- Sextic Surfaces with Ten Triple Points
- Sextic Surfaces with 10 Triple Points
- Topology, Geometry, and Equations of Normal Surface Singularities
Summary
Abstract
This article gives some overview on Singular, a computer algebra system for polynomial computations with special emphasis on the needs of commutative algebra, algebraic geometry and singularity theory, which has been developed under the guidance of G.-M. Greuel, G. Pfister and the second author [31]. We draw the bow from Singular's early years to its latest features. Moreover, we present some explicit calculations, focusing on applications in singularity theory.
Introduction
By the development of effective computer algebra algorithms and of powerful computers, algebraic geometry and singularity theory (like many other disciplines of pure mathematics) have become accessible to experiments. Computer algebra may help
to discover unexpected mathematical evidence, leading to new conjectures or theorems, later proven by traditional means,
to construct interesting objects and determine their structure (in particular, to find counter-examples to conjectures),
to verify negative results such as the non-existence of certain objects with prescribed invariants,
to verify theorems whose proof is reduced to straightforward but tedious calculations,
to solve enumerative problems, and
to create data bases.
In fact, in the last decades, there is a growing number of research articles in algebraic geometry and singularity theory originating from explicit computations (such as [1] and [46] in this volume).
What abilities of a computer algebra system are needed to become a valuable tool for algebraic geometry and, in particular, for singularity theory? First of all, the system needs an efficient representation of polynomials with exact coefficients.
- Type
- Chapter
- Information
- Singularities and Computer Algebra , pp. 247 - 272Publisher: Cambridge University PressPrint publication year: 2006