MA-522 Computer Algebra
Fall 2009
SAS 1220, Tue&Thu 12h00-13h15

Syllabus People Maple Projects Homeworks Reading Grading Academics

Current Announcements

Old Announcements see below.

Peoples' home pages: Erich Kaltofen, Classlist

Maple programs for the course (Maple hints).

Homeworks

  • Homework 1 due Sep 29, 16h59, in my mailbox in SAS 3151.
  • Homework 2 due Oct 29, 16h59, in my mailbox in SAS 3151.
  • Homework 3, a 3-5 page term paper on your presentation.

Projects

Computer Help Resources

Syllabus

Course Outline*

Lecture Topic(s) Notes Book(s)
1. Aug 20 Administrative meeting. First algorithms: modulo n arithmetic.

GG §4.3, GG §20
2. Aug 25 Repeated squaring, RSA
3.mws
GG §4.3, GG §20
3. Aug 27 Extended Euclidean algorithm; Chinese remaindering theorem/algorithm
4.mws
GG §2; §3; §5.4
4. Sep 1 Hermite elimination; analysis of Euclid; Newton and Lagrange interpolation;

GG §3.3; §4.5; §5.2
5. Sep 3 Distribution of primes; use of interpolation/CRA.
[Kaltofen and Villard 2004, p. 112]

6. Sep 8 Rational number recovery; continued fraction approximations of a rational number
[Kaltofen and Rolletschek 1989, Theorem 5.1], KR_ratrec.mpl, KR_ratrec.mws, 4.mws
GG §5.10, §5.11
7. Sep 10 Pollard rho; birthday paradox
Pollard rho code: pollard_rho.mpl, pollard_rho.mws
GG §19.4
8. Sep 15 Talk in joint numeric analysis and symbolic computation seminars
see 173. MAPsncintro.pdf and 174. MAPissacKYZ.pdf linked at BASE/ lectures/ lectures. html# mapgenova
9. Sep 17 Primitive elements modulo p; computing discrete logs via baby-steps/giant steps method and Pollard rho
Teske's paper

10. Sep 22 Maple experiments of Pollard rho; Diffie/Hellman key exchange, el Gamal crypto system
discrete_log.mpl
GG §20.3 and §20.4
11. Sep 24 Fraction-free Gaussian elimination


12. Sep 29 Definition of intergral domain, field of quotients; Euclidean algorithm for polynomials over a field; Sylvester resultants
sylvester.mws, sylvester.txt.
GG §25.2, §25.3 and §6.3
13. Oct 1 Fundamental theorem on subresultants

GG §6.10 and §11.2
14. Oct 6 Unique factorization domains


Thurs-Fri, Oct 8-9 Fall Break, no class
15. Oct 13 Algebraic extension fields; construction of a splitting field.


16. Oct 15 Isomorphism of splitting fields; Galois group; separable and inseparable extensions


Fri, Oct 16, 23h59 Last day to drop the course
17. Oct 20 The Berlekamp/Massey algorithm


18. Oct 22 Norms and traces; the fundamental theorem on symmetric functions
tower_of_fields.mws, tower_of_fields.txt.

19. Oct 27 The ring of algebraic integers; cyclotomic extensions; the infrastructure of finite fields


20. Oct 29 Factoring polynomials over finite fields: the distinct degree and Cantor-Zassenhaus algorithm
factmodp.mws
GG §14
21. Nov 3 CanZas continued


22. Nov 5 The Berlekamp polynomial factoring algorithm; Camion's large primes method

GG §14.8
23. Nov 10 Polynomial ideals; term orders; reduction

GG §21
24. Nov 12 Gröbner bases; Buchberger's algorithm


Fri, Nov 13 ;-) Topic for class presentation must be declared at 17h
25. Nov 17 Michael Nehring presents his results
Link to reference and paper.

26. Nov 19 No class


Mon, Nov 23 Approvals of topics for term papers by me are posted
27. Nov 24 Buchberger's algorithm continued
groebner.mws

Wednesday-Friday, Nov 25-27 Thanksgiving, no class
28. Dec 1 Critical pair/completion paradigm: GCD-free basis construction
[Kaltofen 85, Section 3]

29. Dec 3 Wrap-up; possible presentation


Mon Dec 14, 9-12:00, NEW SAS 4201. Presentations
Requested/assigned times: 10-10:30: ???    10:30-11:00 ???    11:00-11:30 ???    11:30-12:00 ???
* This is a projected list and subject to amendment.

Textbook and Notes

I will be closely following whose sections are marked in the above syllabus by GG.

On-line information: All information on courses that I teach (except individual grades) is now accessible via html browsers, which includes this syllabus. Click on my courses' page of my resume. You can also find information on courses that I have taught in the past.

Grading and General Information

Grading will be done with plus/minus refinement.

There will be three homework assignments of approximately equal weight and one Maple programming projects. At the end of the course, each student will give a 30 minute presentation on material from the book not covered by me. A choice of topics will be provided by me. Class attendance will not be monitored in any way. If you need assistance in any way, please let me know (see also the University's policy).

Academic Standards

Late submissions: All homeworks and projects must be submitted on time. The following penalties are given for (unexcused) late submissions:

If you need assistance in any way, please let me know (see also the University's policy).
Swine-flu notice: If you are ill with symptoms of H1N1 influenza (i.e., fever over 100, sore throat, cough, stuffy or runny nose, fatigue, headache, body aches, vomiting and diarrhea) please do not come to class. Instead, immediately contact your medical provider or Student Health Services (515-7107) for advice or to arrange an appointment.
If you are diagnosed with H1N1, please inform your instructor immediately. You will be required to be isolated away from class until at least 24 hours after you are free of fever (100 degrees), or signs of a fever, without the use of fever-reducing medications.

Old Announcements

©2009 Erich Kaltofen. Permission to use provided that copyright notice is not removed.