MA-693/893 Computer Algebra 2 Spring 2022 Reading Course

Current Announcements

• NEW There is a weekly reading assignment, and a bi-weekly homework assignment. I will hold office hours twice/week when questions about the reading and homework are discussed, and when I may ask you questions about what you have read.
• NEW For students planning to take the Computer Algebra Qualifying Exam in Summer 2022, there may be additional homework assignments.
Old Announcements see below.

 Peoples' home pages: Erich Kaltofen, Classlist Maple programs for the course (Maple hints). Homework 1 due ??? on moodle or by email. Homework 2 due ??? on moodle or by email. Homework 3 due ??? on moodle or by email. Homework 4 due ??? on moodle or by email. Homework 5 due ??? on moodle or by email. Homework 6 due ??? on moodle or by email. Projects Computer Help Resources

Course Outline*

Lecture Topic(s) Notes Book(s)
1. Jan 11 Administrative meeting. Strassen's matrix multiplication

GG §12.1
2. Jan 16 Algebraic complexity; analysis of Strassen's scheme

4. Jan 22 LUP factorization < matrix multiplication (Bunch-Hopcroft algorithm)

5. Jan 25 Outline of black box linear algebra

6. Jan 29 Hensel lifting

GG §15.4
7. Feb 1 Dixon's algorithm
Numer. Math., vol. 40, nr. 1 (1982)
8. Feb 5 P-adic numbers; recap of black box linear algebra

GG §9.6
9. Feb 8 the Berlekamp-Massey algorithm
Section 2.2 in JSC vol. 36 (2003)
GG §12.3
10. Feb 12 Proof of Wiedemann algorithm via the Schwartz/Zippel lemma
Section 3 in Math. Comput. vol. 64 (1995)
GG §6.9, Lemma 6.44
11. Feb 15 Wiedemann algorithm for singular matrices; the use of preconditioners
See talk at Fq6
Further reading LAA vol. 343-344 (2002)
GG §12.4
12. Feb 19 Gauss's lemma; GCD of several multivariate polynomials via linear combinations
Theorem 6.2 in J. ACM, vol. 35 (1988)
GG §6.2
13. Feb 22 GCD of several multivariate polynomials via linear combinations continued

14. Feb 27 Integration of rational functions
My Lect. Notes
GG §22
15. Mar 1

Week of Mar 5-9 Spring Break, no class
16. Mar 12 Lattice basis reduction (LLL)

GG §12.1-12.3
17. Mar 15 LLL continued

18. Mar 19 Integration of rational functions continued; proof that log(x) is irrational; Rothstein-Trager theorem

Wed, Mar 21, 5pm Last day to drop the course
19. Mar 22 GGH public key crypto-system; knapsack based crypto
Section 4 in JSC vol. 29, pp. 891-919 (2000)

20. Mar 26 Differential fields; quotient fields; algebraic function fields; proof that log(x) is transcendental
My Lect. Notes

21. Mar 29 Elementary Liouville extensions; structure theorem (not proved)

22. Apr 2 Liouville's theorem (not proved); impossibilities of closed form solution of the integral of certain transcendental functions

Tue, Apr 3, 5pm
Papers for class presentation must be declared at 5pm
23. Apr 5 Semi-algebraic sets; the principle of quantifier elimination on examples
Section 2 in JSC vol. 29, pp. 891-919 (2000)

24. Apr 9 Proof of Sturm's theorem; Cauchy root bound

GG Exercise 4.32
Tue, Apr 10
Approvals of papers for presentation by me are posted
25. Apr 12 Cauchy principle of argument; Cauchy index of a real rational function on an interval
Handout: chapters in Marden's and Gantmacher's books

26. Apr 16 Routh-Hurwitz algorithm for complex root isolation

27. Apr 19 Seidenberg's method
Handout: Seidenberg's paper; Section 6 in AAECC vol. 1, nr. 2, pp. 135-148 (1990)

Sat, Apr 21 East Coast Computer Algebra Day, Washington College
28. Apr 23 Paper presentations begin; recap of Fall-Spring courses

29. Apr 26 Extra topic: FFT

GG §8.2
??? Presentations
Friday, May 6, 4:59pm Spring 2022 grades due
* 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.