- normal forms and standard bases of ideals and modules,
- syzygies, free resolutions and the proof of the Buchberger criterion,
- computation of normalization of Noetherian rings,
- computation of primary decomposition of ideals,
- Hilbert function,
- Ext and Tor.

# Module MAT-41-11-M-7

## Computer Algebra (M, 9.0 LP)

## Module Identification

Module Number | Module Name | CP (Effort) |
---|---|---|

MAT-41-11-M-7 | Computer Algebra | 9.0 CP (270 h) |

## Basedata

CP, Effort | 9.0 CP = 270 h |
---|---|

Position of the semester | 1 Sem. irreg. SuSe |

Level | [7] Master (Advanced) |

Language | [EN] English |

Module Manager | |

Lecturers | |

Area of study | [MAT-AGCA] Algebra, Geometry and Computer Algebra |

Reference course of study | [MAT-88.105-SG] M.Sc. Mathematics |

Livecycle-State | [NORM] Active |

## Courses

Type/SWS | Course Number | Title | Choice in Module-Part | Presence-Time / Self-Study | SL | SL is required for exa. | PL | CP | Sem. | |
---|---|---|---|---|---|---|---|---|---|---|

4V+2U | MAT-41-11-K-7 | Computer Algebra
| P | 84 h | 186 h | - | - | PL1 | 9.0 | irreg. SuSe |

- About [MAT-41-11-K-7]: Title: "Computer Algebra"; Presence-Time: 84 h; Self-Study: 186 h

## Examination achievement PL1

- Form of examination: oral examination (20-30 Min.)
- Examination Frequency: each semester
- Examination number: 86170 ("Computer Algebra")

## Evaluation of grades

The grade of the module examination is also the module grade.

## Contents

## Competencies / intended learning achievements

Upon successful completion of this module, the students have gained advanced knowledge in the theory of computer algebra. They know how problems from commutative algebra, algebraic geometry and their practical applications translate to a computer algebra system and how they can be solved by using algorithms. Moreover, they are able to program advanced algorithms of computer algebra. They understand the proofs presented in the lecture and are able to comprehend and explain them.

By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies independently or by team work.

## Literature

- G.-M. Greuel, G. Pfister: A SINGULAR Introduction to Commutative Algebra,
- W. Decker, C. Lossen: Computing in Algebraic Geometry - A Quick Start using Singular,
- W. Decker, F.-O. Schreyer: Varieties, Gröbner bases, and algebraic curves,
- W. Decker, G. Pfister: A First Course in Computational Algebraic Geometry,
- D. Cox, J. Little, D. O'Shea: Ideals, Varieties, and Algorithms.

## Requirements for attendance of the module (informal)

#### Modules:

- [MAT-10-1-M-2] Fundamentals of Mathematics (M, 28.0 LP)
- [MAT-40-11-M-4] Commutative Algebra (M, 9.0 LP)

#### Courses

- [MAT-12-11-K-2] Algebraic Structures (2V+2U, 5.5 LP)
- [MAT-14-12-K-3] Introduction to Symbolic Computing (4V+2U, 9.0 LP)

## Requirements for attendance of the module (formal)

**None**

## References to Module / Module Number [MAT-41-11-M-7]

Course of Study | Section | Choice/Obligation |
---|---|---|

[INF-88.79-SG] M.Sc. Computer Science | [Specialisation] Specialization 1 | [WP] Compulsory Elective |

Module-Pool | Name | |

[MAT-41-MPOOL-7] | Specialisation Algebraic Geometry and Computer Algebra (M.Sc.) | |

[MAT-AM-MPOOL-7] | Applied Mathematics (Advanced Modules M.Sc.) | |

[MAT-RM-MPOOL-7] | Pure Mathematics (Advanced Modules M.Sc.) |