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Course Goals |
Introducing the algorithms for the part of the commutative algebra which is suitable for computer calculations and encouraging students to implement them on their own using a computer algebra system (ApCoCoA). |
Prerequisite(s) |
. Linear Algebra I
. Algebra I |
Corequisite(s) |
- |
Special Requisite(s) |
The minimum qualifications that are expected from the students who want to attend the course.(Examples: Foreign language level, attendance, known theoretical pre-qualifications, etc.) |
Instructor(s) |
Assist. Prof. Dr. Levent Çuhacı |
Course Assistant(s) |
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Schedule |
Day, hours, XXX Campus, classroom number. |
Office Hour(s) |
Instructor name, day, hours, XXX Campus, office number. |
Teaching Methods and Techniques |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
Principle Sources |
Computational Commutative Algebra 1 –Martin Kreuzer, Lorenzo Robbiano |
Other Sources |
Computer, ApCoCoA (free computer algebra software) |
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Course Schedules |
Week |
Contents |
Learning Methods |
1. Week |
Polynomial rings and extended Euclidean algorithm |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
2. Week |
Monomial ideals and term orderings |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
3. Week |
Macaulay’s basis theorem and Dickson’s Lemma |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
4. Week |
Grading of rings and homogenous ideals |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
5. Week |
Syzygy computations and lifting of syzygies |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
6. Week |
Buchberger Algorithm and computing Gröbner bases of ideals |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
7. Week |
Computation of syzygy modules by the help of Gröbner bases |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
8. Week |
Computation of the kern and images of homomorphims of modules and lifting of linear maps |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
9. Week |
Computation of elimination, localization and saturation |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
10. Week |
Systems of polynomial equations |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
11. Week |
Some applications of Gröbner Bases |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
12. Week |
Some examples |
Explaining the theoretical background in class with computation methods in an algorithmic way then encouraging the students to implement them (see Worksheet) |
13. Week |
Reviewing some topics for the Exam |
Worksheet |
14. Week |
Reviewing some topics for the Exam |
Worksheet |
15. Week |
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16. Week |
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17. Week |
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Program Outcomes |
PO-1 | Have scientific research in mathematics and computer science in the level of theoretical and practical knowledge. | PO-2 | On the basis of undergraduate level qualifications, develop and deepen the same or a different areas of information at the level of expertise, and analyze and interpret by using statistical methods | PO-3 | Develop new strategic approaches for the solution of complex problems encountered in applications related to the field and unforeseen and take responsibility for the solution. | PO-4 | Evaluate critically skills acquired in the field of information in the level of expertise and assess the learning guides. | PO-5 | Transfer current developments in the field and their work to the groups inside and outside the area supporting with quantitative and qualitative datas as written, verbal and visual by a systematic way. | PO-6 | Use information and communication technologies with computer software in advanced level. | PO-7 | Develop efficient algorithms by modeling problems faced in the field and solve such problems by using actual programming languages. | PO-8 | Respect to social, scientific, cultural and ethical values at the stages of data collection related to the field, interpretation, and implementation. | PO-9 | To solve problems related to the field, establish functional interacts by using strategic decision making processes. | PO-10 | Establish and discuss in written, oral and visual communication in an advanced level by using at least one foreign language. |
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Learning Outcomes |
LO-1 | I. Understand polynomial rings and monomial ideals. | LO-2 | II. Undestand the theoritical background and they can compute Gröbner basis and the reduced Gröbner basis of an ideal. | LO-3 | III. Understand the theoretical background of syzygies, colon ideals, localization and saturation and they can compute them. | LO-4 | IV. Apply these techniques to solve polynomial equation systems. | LO-5 | V. Implement these techniques by using a computer algebra program, for example ApCoCoA. |
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