

MA693/893
Computer Algebra 2 Spring 2022 Reading Course 
Syllabus  People  Maple  Projects  Homeworks  Reading  Grading  Academics 
Current Announcements

Peoples' home pages:
Erich Kaltofen,
Maple programs for the course (Maple hints).


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




3. Jan 18 
Winograd's scheme



4. Jan 22 
LUP factorization < matrix multiplication
(BunchHopcroft 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 
Padic numbers;
recap of black box linear algebra


GG §9.6


9. Feb 8 
the BerlekampMassey 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. 343344 (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 59  Spring Break, no class  
16. Mar 12 
Lattice basis reduction (LLL)


GG §12.112.3


17. Mar 15 
LLL continued




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




Wed, Mar 21, 5pm  Last day to drop the course 

19. Mar 22 
GGH public key cryptosystem;
knapsack based crypto

Section 4 in
JSC vol. 29, pp. 891919 (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 
Semialgebraic sets;
the principle of quantifier elimination
on examples

Section 2 in
JSC vol. 29, pp. 891919 (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 
RouthHurwitz algorithm for
complex root isolation




27. Apr 19 
Seidenberg's method

Handout: Seidenberg's paper;
Section 6 in
AAECC vol. 1, nr. 2, pp. 135148 (1990)



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


29. Apr 26 
Extra topic: FFT

GG §8.2


???  Presentations  
Friday, May 6, 4:59pm  Spring 2022 grades due 
Online 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.
There will be six homework assignments of approximately equal weight. At the end of the course, each student will give a 30 minute presentation and write a 35 page discussion of the ideas, or write a program implementing the algorithms in what you have read. I expect that you attend my office hours. If you need assistance in any way, please let me know (see also the University's policy).
©2022 Erich Kaltofen. Permission to use provided that copyright notice is not removed.