Prerequisite Courses:
Course Language:
English
Course Objectives:
The aim of this course is to provide students with a knowledge and understanding of basic results of computational algebra and basic algorithmic approaches to algebra and their implementations.
Course Content:
The Elimination and Extension Theorems, The Geometry of Elimination, Implicitization, Singular Points and Envelopes, Gröbner Bases and the Extension Theorem, Hilbert's Nullstellensatz, Radical Ideals and the Ideal-Variety Correspondence, Sums, Products and Intersections of Ideals, Zariski Closure and Quotients of Ideals, Irreducible Varieties, Decomposition of a Variety, Polynomial Mappings, Quotients of Polynomial Rings, Algorithmic Computations in k[x1, . . . , xn]/I, The Coordinate Ring of an Affine Variety, Primary Decomposition of Ideals, The Variety of a Monomial Ideal.
Course Methodology:
1: Lecture, 2: Problem Solving
Course Evaluation Methods:
A: Written examination, B: Homework
Vertical Tabs
Course Learning Outcomes
| Learning Outcomes | Program Learning Outcomes | Teaching Methods | Assessment Methods |
| 1) Knows about algebraic geometry and commutative algebra. | 1,2,3 | 1,2 | A,B |
| 2) Knows about Gröbner basis theory and understands the relevant algorithms and their analysis. | 1,2,3 | 1,2 | A,B |
| 3) Knows how to use computational algorithms in commutative algebra and Gröbner basis theory to solve various problems. | 1,2,3 | 1,2 | A,B |
| 4) Participates in research and scientific discussions and is able to learn new topics in computational algebra. | 1,2,3 | 1,2 | A,B |
Course Flow
| Week | Topics | Study Materials |
| 1 | The Elimination and Extension Theorems, The Geometry of Elimination | Textbook 3.1, 3.2 |
| 2 | Implicitization | Textbook 3.3 |
| 3 | Singular Points and Envelopes | Textbook 3.4 |
| 4 | Gröbner Bases and the Extension Theorem | Textbook 3.5 |
| 5 | Hilbert's Nullstellensatz | Textbook 4.1 |
| 6 | Radical Ideals and the Ideal-Variety Correspondence | Textbook 4.2 |
| 7 | Sums, Products and Intersections of Ideals | Textbook 4.3 |
| 8 | Zariski Closure and Quotients of Ideals | Textbook 4.4 |
| 9 | Irreducible Varieties, Decomposition of a Variety | Textbook 4.5 |
| 10 | Decomposition of a Variety into Irreducibles | Textbook 4.6 |
| 11 | Primary Decomposition of Ideals | Textbook 4.8 |
| 12 | Polynomial Mappings, Quotients of Polynomial Rings | Textbook 5.1, 5.2 |
| 13 | Algorithmic Computations in k[x1, . . . , xn]/I | Textbook 5.3 |
| 14 | The Coordinate Ring of an Affine Variety | Textbook 5.4 |
Recommended Sources
| Textbook | D. A. Cox, J. Little, D. O’Shea, Ideals, Varieties, and Algorithms, Springer, Fourth Edition, 2015. |
| Additional Resources |
G.-M. Greuel, G. Pfister, A Singular introduction to commutative algebra, Springer, 2002.
D. Shafer, V. Romanovski, The center and cyclicity problems: a computational algebra approach, Birkhäuser Basel, 2009. V.G. Romanovski, M. Presern, An approach to solving systems of polynomials via modular arithmetics with applications, Journal of Computational and Applied Mathematics, 236, 196–208, 2011. |
Material Sharing
| Documents | |
| Assignments | |
| Exams |
Assessment
| IN-TERM STUDIES | NUMBER | PERCENTAGE |
| Mid-terms | 1 | 70 |
| Quizzes | 0 | 0 |
| Assignments | 5 | 30 |
| Total | 100 | |
| CONTRIBUTION OF FINAL EXAMINATION TO OVERALL GRADE | 1 | 50 |
| CONTRIBUTION OF IN-TERM STUDIES TO OVERALL GRADE | 50 | |
| Total | 100 |
Course’s Contribution to Program
| COURSE'S CONTRIBUTION TO PROGRAM | ||||||
| No | Program Learning Outcomes | Contribution | ||||
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | The ability to make computation on the basic topics of mathematics such as limit, derivative, integral, logic, linear algebra and discrete mathematics which provide a basis for the fundamental research fields in mathematics (i.e., analysis, algebra, differential equations and geometry) | x | ||||
| 2 | Acquiring fundamental knowledge on fundamental research fields in mathematics | x | ||||
| 3 | Ability form and interpret the relations between research topics in mathematics | x | ||||
| 4 | Ability to define, formulate and solve mathematical problems | x | ||||
| 5 | Consciousness of professional ethics and responsibility | x | ||||
| 6 | Ability to communicate actively | x | ||||
| 7 | Ability of self-development in fields of interest | x | ||||
| 8 | Ability to learn, choose and use necessary information technologies | x | ||||
| 9 | Lifelong education | x | ||||
ECTS
| ECTS ALLOCATED BASED ON STUDENT WORKLOAD BY THE COURSE DESCRIPTION | |||
| Activities | Quantity |
Duration (Hour) |
Total Workload (Hour) |
| Course Duration (14x Total course hours) | 14 | 3 | 42 |
| Hours for off-the-classroom study (Pre-study, practice) | 14 | 5 | 70 |
| Mid-terms (Including self study) | 1 | 10 | 10 |
| Quizzes | |||
| Assignments | 5 | 5 | 25 |
| Final examination (Including self study) | 1 | 20 | 20 |
| Total Work Load | 167 | ||
| Total Work Load / 25 (h) | 6.68 | ||
| ECTS Credit of the Course | 7 | ||