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The Computer Algebra System OSCAR

Algorithms and Examples

  • Book
  • © 2025

Overview

Editors:
  1. Wolfram Decker

    1. Department of Mathematics, RPTU Kaiserslautern-Landau, Kaiserslautern, Germany

  2. Christian Eder

    1. Department of Mathematics, RPTU Kaiserslautern-Landau, Kaiserslautern, Germany

  3. Claus Fieker

    1. Department of Mathematics, RPTU Kaiserslautern-Landau, Kaiserslautern, Germany

  4. Max Horn

    1. Department of Mathematics, RPTU Kaiserslautern-Landau, Kaiserslautern, Germany

  5. Michael Joswig

    1. Institute of Mathematics, Technical University of Berlin, Berlin, Germany

  • A unique resource on the new computer algebra system OSCAR
  • Discusses the interaction of computational tools from different disciplines in algebra, geometry, and number theory
  • Presents explicit OSCAR code written by experts

Part of the book series: Algorithms and Computation in Mathematics (AACIM, volume 32)

  • 13k Accesses

  • 12 Citations

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About this book

This book presents version 1.0 of the new Computer Algebra System OSCAR.

Written in Julia, OSCAR builds on and vastly extends four cornerstone systems:

ANTIC for number theory, GAP for group and representation theory,  polymake for polyhedral and tropical geometry, and  Singular for commutative algebra and algebraic geometry.

It offers powerful computational tools that transcend the boundaries of the individual disciplines involved.

It is freely available, open source software.

 

The book is an invitation to use OSCAR.  With discussions of theoretical and algorithmic aspects included, it offers a multitude of explicit code snippets.  These are valuable for interested researchers from graduate students through established experts.

 

 

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Table of contents (19 chapters)

  1. Front Matter

    Pages i-xviii
    Download chapter PDF

  2. OSCAR

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. What Is OSCAR?

      • Claus Fieker, Michael Joswig
      Pages 3-17

  3. The Cornerstones

    1. Front Matter

      Pages 19-19
      Download chapter PDF

    2. Commutative Algebra and Algebraic Geometry

      • Janko Böhm, Wolfram Decker, Frank-Olaf Schreyer
      Pages 21-79

    3. Number Theory

      • Claus Fieker, Tommy Hofmann
      Pages 81-105

    4. Group Theory

      • Max Horn, Claus Fieker
      Pages 107-138

    5. Polyhedral Geometry

      • Taylor Brysiewicz, Michael Joswig
      Pages 139-164

  4. Applications

    1. Front Matter

      Pages 165-165
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    2. Algorithms for Gromov–Witten Invariants of Elliptic Curves

      • Janko Böhm, Firoozeh Dastur, Alain Hoffmann, Hannah Markwig, Ali Traore
      Pages 167-190

    3. Toric Geometry

      • Martin Bies, Lars Kastner
      Pages 191-211

    4. Algorithms for GIT-Fans of Affine Torus Actions

      • Janko Böhm, Thomas Breuer
      Pages 213-224

    5. Elliptic Fibrations on Vinberg’s Most Algebraic K3 Surface

      • Simon Brandhorst, Matthias Zach
      Pages 225-241

    6. An Atlas of Orthogonal Representations

      • Thomas Breuer, Gabriele Nebe, Richard Parker
      Pages 243-270

    7. Gröbner Bases for Polynomial Ideals and Applications

      • Christian Eder, Rafael Mohr, Mohab Safey El Din
      Pages 271-295

    8. Invariant Theory

      • Wolfram Decker, Lakshmi Ramesh, Johannes Schmitt
      Pages 297-331

    9. Monomial Bases in Lie Theory

      • Xin Fang, Ghislain Fourier
      Pages 333-349

    10. Matroids

      • Daniel Corey, Lukas Kühne, Benjamin Schröter
      Pages 351-368

    11. Generic Root Counts of Tropically Transverse Systems an Invitation to Tropical Geometry in OSCAR

      • Isaac Holt, Yue Ren
      Pages 369-401

    12. Faithful Tropicalization of Hyperelliptic Curves

      • Hannah Markwig, Lukas Ristau, Victoria Schleis
      Pages 403-428
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Editors and Affiliations

  • Department of Mathematics, RPTU Kaiserslautern-Landau, Kaiserslautern, Germany

    Wolfram Decker, Christian Eder, Claus Fieker, Max Horn

  • Institute of Mathematics, Technical University of Berlin, Berlin, Germany

    Michael Joswig

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Bibliographic Information

  • Book Title: The Computer Algebra System OSCAR

  • Book Subtitle: Algorithms and Examples

  • Editors: Wolfram Decker, Christian Eder, Claus Fieker, Max Horn, Michael Joswig

  • Series Title: Algorithms and Computation in Mathematics

  • DOI: https://doi.org/10.1007/978-3-031-62127-7

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2025

  • Hardcover ISBN: 978-3-031-62126-0Published: 31 January 2025

  • Softcover ISBN: 978-3-031-62129-1Due: 14 February 2026

  • eBook ISBN: 978-3-031-62127-7Published: 30 January 2025

  • Series ISSN: 1431-1550

  • Series E-ISSN: 2512-3254

  • Edition Number: 1

  • Number of Pages: XVIII, 491

  • Number of Illustrations: 56 b/w illustrations

  • Topics: Algebra, Symbolic and Algebraic Manipulation

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