Ideals, Varieties, Algorithms(MAT473)
| Course Code | Course Name | Semester | Theory | Practice | Lab | Credit | ECTS |
|---|---|---|---|---|---|---|---|
| MAT473 | Ideals, Varieties, Algorithms | 7 | 3 | 0 | 0 | 3 | 5 |
| Prerequisites | |
| Admission Requirements |
| Language of Instruction | French |
| Course Type | Elective |
| Course Level | Bachelor Degree |
| Course Instructor(s) | Meral TOSUN mtosun@gsu.edu.tr (Email) |
| Assistant | |
| Objective | The purpose of this course is to learn about Groebner basis which is useful to solve some problems on algebraic varieties, especially for the solution of systems of equations, to understand how to use it in the proof of theorem extension. |
| Content |
Ring theory and fields (summary), Polynomial rings and affine space, Affine varieties, Parametrization, Ideals, One variable polynomials; Monomial orders, Division algorithm, Monomial ideals and Dickson's lemma, Hilbert bases theorem, Groebner bases, Properties of Groebner bases, Buchbergers algorithm, Applicaitons of Groebner bases; Elimination and Extension theorems, Resultants and the extension theorem. |
| Course Learning Outcomes |
To know how to calculate a Groebner base for an ideal by using the algorithm of Buchberger To know how to use the Groebner bases to solve the appearence problem for an ideal To be able to use the Groebner bases in the theory of elimination to solve the system of equations |
| Teaching and Learning Methods | Lecture, discussion, problem solving |
| References | Ideals, Varieties and Algorithms, D. Cox, J. Little, D. O’Shea. |
Theory Topics
| Week | Weekly Contents |
|---|---|
| 1 | Ring theory and fields (summary) |
| 2 | Polynomial rings and affine space, Affine varieties |
| 3 | Ideals, One variable polynomials |
| 4 | Monomial orders, Division algorithm |
| 5 | Monomial ideals and Dickson's lemma |
| 6 | Hilbert bases theorem and Groebner bases |
| 7 | Preparation for the midterm examination |
| 8 | Midterm Examination |
| 9 | Properties of Groebner bases |
| 10 | Buchbergers algorithm |
| 11 | Elimination and Extension theorems |
| 12 | Unique factorization and resultants |
| 13 | Resultants and the extension theorem |
| 14 | Preparation for the final exam |
Practice Topics
| Week | Weekly Contents |
|---|
Contribution to Overall Grade
| Number | Contribution | |
|---|---|---|
| Contribution of in-term studies to overall grade | 4 | 60 |
| Contribution of final exam to overall grade | 1 | 40 |
| Toplam | 5 | 100 |
In-Term Studies
| Number | Contribution | |
|---|---|---|
| Assignments | 2 | 20 |
| Presentation | 2 | 20 |
| Midterm Examinations (including preparation) | 1 | 20 |
| Toplam | 5 | 60 |
| No | Program Learning Outcomes | Contribution | ||||
|---|---|---|---|---|---|---|
| 1 | 2 | 3 | 4 | 5 | ||
| 1 | understands principles of deductive reasoning; has experience to verify well-foundedness and exactness of mathematical statements in systematic ways; | X | ||||
| 2 | can properly state and use concepts and results of major mathematical interest; | X | ||||
| 3 | masters current computational techniques and algorithms; has a good ability in their use; can identify relevant tools, among those one has learned, suitable to solve a problem and is able to judge whether or not one is in possession of these tools; | X | ||||
| 4 | is able to express one’s mathematical ideas in an organised way both in written and oral forms; | X | ||||
| 5 | understands relations connecting substantial concepts and results; can switch from one viewpoint to another on mathematical objects (pictures, formulae, precise statements, heuristic trials, list of examples,...); | X | ||||
| 6 | has followed individually a guided learning strategy; has pursued steps toward the resolution of unfamiliar problems; | X | ||||
| 7 | has a theoretical and practical knowledge in computer science well adapted for learning a programming language; | |||||
| 8 | has investigated the relevance of modeling and using mathematical tools in natural sciences and in the professional life; is conscious about historical development of mathematical notions; | X | ||||
| 9 | has followed introduction to some mathematical or non-mathematical disciplines after one’s proper choice; had experience to learn selected subjects according to one’s proper arrangement; | X | ||||
| 10 | masters French language as well as other foreign languages, to a level sufficient to study or work abroad. | X | ||||
| Activities | Number | Period | Total Workload |
|---|---|---|---|
| Total Workload | 0 | ||
| Total Workload / 25 | 0.00 | ||
| Credits ECTS | 0 | ||