Computer algebra, which is also called formula manipulation, treats each mathematical expression as a symbolic sequence, and
perform some mathematical operations such as arithmetics, expansions/factorizations of polynomials, and even differentiations
and integrations exactly in mathematical formulas. The central objects dealt by computer algebra are numbers and polynomials,
and there are specific algorithms such as finding common factors of plural polynomials. At first, some of these algorithms
will be talked as an introduction of computer algebra. After that, the concept of Groebner basis, which plays an important
role in solving a system of algebraic equations, ideal membership problem, and so on, will be talked and some central topics
of modern computer algebraic research will be discussed.
- To recognize that calculation by computer is not limited to numerical calculation.
- To be able to calculate the GCD and to understand the relation between the resultant and the GCD.
- To understand the basics of computer algebraic treatment of polynomial factorization.
| Class schedule | HW assignments (Including preparation and review of the class.) | Amount of Time Required | |
|---|---|---|---|
| 1. | Introduction to computer algebra systems. Review of algebra (1) : the ring, the field and the Ideal. |
Research on computational algebra systems in advance. | 90minutes |
| Review of the subjects related to algebra. | 100minutes | ||
| 2. | Review of algebra (2) : the polynomial ring. | Review of the subjects related to algebra. | 190minutes |
| 3. | Eucledian algorithm | Review the last lecture. | 190minutes |
| 4. | Extended Eucledian algorithm, the pseudo remainder sequence. | Review the last lecture. | 190minutes |
| 5. | Polynomial remainder sequence (PRS) and some improved PRS algorithms. | Review the last lecture. | 190minutes |
| 6. | Resultant. | Review the last lecture. | 190minutes |
| 7. | Subresultant and the relation between the subresultant and the GCD. | Review the last lecture. | 190minutes |
| 8. | Modular algorithm (1): Chinese remainder theorem. | Review the last lecture. | 190minutes |
| 9. | Modular algorithm (2): reconstruction of rational numbers and polynomials. | Review the last lecture. | 190minutes |
| 10. | Modular algorithm (3): Hensel lifting. |
Review the last lecture. | 190minutes |
| 11. | Factorization of polynomial (1): square free decomposition. | Review the last lecture. | 190minutes |
| 12. | Factorization of polynomial (2): one variable polynomial over finite field. | Review the last lecture. | 190minutes |
| 13. | The fundamentals of Groebner basis. | Review the last lecture. | 190minutes |
| 14. | Term-end examination and its review. | Review the total lectures. | 190minutes |
| Total. | - | - | 2660minutes |
| Term-end examination | Regular assignments | Total. | |
|---|---|---|---|
| 1. | 0% | 10% | 10% |
| 2. | 20% | 20% | 40% |
| 3. | 30% | 20% | 50% |
| Total. | 50% | 50% | - |
Term-end examination (50%) and two or three regular assignments (50%).
Submit each examination/report in the specified format on time. Complete the assigned tasks without fail (correct results for calculations, and correct deductions for proofs). When a discussion is required, write a discussion, not an impression. The answer/reports should be written with the reader in mind (in principle, the process should be written as well as the answer). You will get 100 points if you have done these things perfectly. You will be given a halfway point for each report, so if you achieve 60% of the above, you will pass.
As described above, the term-end examination (written test) will be 50% and the regular assignments (reports) 50% in the evaluation. But, if it becomes difficult to give the term-end examination in the classroom, the final assignment (report; it may be divided into two or three assignments) will be substituted for the term-end examination (to be announced at the beginning of the class or when the situation suddenly changes).
Submit each examination/report in the specified format on time. Complete the assigned tasks without fail (correct results for calculations, and correct deductions for proofs). When a discussion is required, write a discussion, not an impression. The answer/reports should be written with the reader in mind (in principle, the process should be written as well as the answer). You will get 100 points if you have done these things perfectly. You will be given a halfway point for each report, so if you achieve 60% of the above, you will pass.
As described above, the term-end examination (written test) will be 50% and the regular assignments (reports) 50% in the evaluation. But, if it becomes difficult to give the term-end examination in the classroom, the final assignment (report; it may be divided into two or three assignments) will be substituted for the term-end examination (to be announced at the beginning of the class or when the situation suddenly changes).
| ways of feedback | specific contents about "Other" |
|---|---|
| Feedback in the class |
No textbook.
Reference: "Fundamentals of Computer Algebra," K. Nagasaka and H Iwane (edt.), T. Kitamoto et al, Kyoritsy pub., 2019 (in Japanese).
Reference: "Fundamentals of Computer Algebra," K. Nagasaka and H Iwane (edt.), T. Kitamoto et al, Kyoritsy pub., 2019 (in Japanese).
Students are expected to have taken subjects related to algebraic, e.g., "Fundamental Algebra", "Algebra 1", and so on.
- Course that cultivates an ability for utilizing knowledge
| Work experience | Work experience and relevance to the course content if applicable |
|---|---|
| N/A | N/A |
Last modified : Sat Sep 09 07:28:21 JST 2023