What's New
[8/6 updated] All lectures were finished. Thank you for your cooperation.
Course Info.
- Course Title: Algorithms for polynomial systems: elimination and Groebner basis
- Instructor: Xavier Dahan (Faculty of Mathematics / MI Assistant Professor)
- TA: Shun'ichi Yokoyama (Graduate School of Mathematics / GCOE Super Research Assistant)
- Time and Place: Thursday 10:30am -- 12:00noon in Middle Seminar Room #2
Useful Software Links
If you're looking for a good link to install something, you might find one here. If you find a useful link somewhere, let us know and we'll post it.
- Mathematica Download Page (Virtual Office, Not accessible from outside)
Schedule and Lectures
Apr. 15, 2010 / #1 Introduction & Motivation
Apr. 22, 2010 / #2 Univariate Polynomial's GCD: Bezout Identity
- Lecture Sildes II [5/6: Latest Edit]
- TA's Handout 2: Vocabulary 1
- Mathematica: Polynomial Extended GCD
May. 6, 2010 / #3 About Univariate Polynomial and intro. to Multivariate
- Lecture Sildes III [5/13: Latest Edit]
- TA's Handout 3: Vocabulary 2
May. 13, 2010 / #4 About Multivariate Polynomial and Division Algorithm
- #3 Continued...
May. 20, 2010 / #5 About unicity and Introduction to Groebner basis I
- Lecture Sildes IV [5/27: Latest Edit]
- Practice Test I: Monomial Order and Division Algorithm
- TA's Handout 4: Easy Exercises
May. 27, 2010 / #6 Introduction to Groebner basis II
- TA's Handout Extra: Remarks for P.T. I in Japanese [Indexed 6/3]
- Practice Test II: Division equality and monomial ideals
Jun. 3, 2010 / #7 Buchberger's algorithm
Jun. 10, 2010 / #8 Syzygies and Overview of Resultants
- #7 Continued...
Jun. 17, 2010 / #9 Resultants I
- Lecture Sildes VI
- [NEW!!] Practice Test III: Around the Buchberger algorithm
- 6/29 modified: Here is the CORRECTED version, especially the following.
- CORRECTION 1: Ex.5-Q2. "Next, select the pair (1,3)" → NO! It is (2,3).
- CORRECTION 2: Ex.5-Q3. "Next the pair (2,3)" → NO! It is (1,3).
- CORRECTION 3: It is written "There are 2 pairs above [1,l] and [3,l] such that Test 2 works" but it doesn't work. However, it is not so serious because actually we have S(f1,f3) --> 0 so the other questions are correct.
- CORRECTION 4: Ex.6-Q3, It is not correct, but there is a homogeneous syzygy of multi-degree X^\gamma, so it is not "too incorrect".
- For students who have difficulties, it is highly recommended to come to the office hours, Monday from 2pm to 4pm.
Jun. 24, 2010 / #10 Resultants II + Elimination and Nullstellensatz I
- [6/29 uploaded] Resultant and Applications
- [6/29 uploaded] TA's Handout 5: Examples and Remarks
- Mathematica File: Syl-2.nb
- Sylvester matrix, intersection of 2 plane curves, Eulidean algorithm for the resultant
- Mathematica File: VanishOnAlgNbr.nb
- Contains 3 examples of computation of a vanishing polynomial of some algebraic numbers
Jul. 1, 2010 / #11 Elimination and Nullstellensatz II
Jul. 8, 2010 / #12 Elimination and Nullstellensatz III
Jul. 15, 2010 / #13 FINAL LECTURE
- Files that are useful to solve Test V. Please download if you need.