Within the scope of the "Joint Advanced Student School
- JASS'2007" in St. Petersburg (March
25 - April 4), we offer the course "Polynomials:
Their Power and How to Use Them".
Directors:
Prof. Dr. Yu. V. Matiyasevich (Steklov
Institute, St. Petersburg)
Prof. Dr. E. W. Mayr (TU München)
Assistant:
Application:
The course addresses highly motivated and interested students. Active preparation
and contribution of the participants will be required. The course language
is English. Expenses for travel, board and lodging of german participants
are covered by the school.
Application deadline for GERMAN applicants: January
29, 2007.
Application form: PDF
.
German applicants should send their applications
to D. Chibisov.
Russian
applicants should send their applications to Prof.
Dr. Yu. V. Matiyasevich.
Organization:
Every participant is expected to give a talk and
to prepare a paper on the topic of his/her talk. Some help is provided on
the organizational stuff page
Course Description:
Constructions in polynomial
rings with several variables and coefficients from a field like the rationals
or GF(2) play an important role in mathematics, computer science, and engineering.
Especially algorithmic questions about polynomials, their roots, and polynomial
ideals is an important issue for various applications in cryptography, geometry,
robotics, etc. The course is devoted, besides the fundamentals of algorithmic
algebra and algorithmic number theory, to decidability and complexity questions
in an algebraic setting as well as to recent applications of algorithmic
algebra in mathematics and computer science.
| Talks |
|
| Maximilian Butz Basics about polynomials: arithmetics with polynomials; complex and real roots of univariate polynomials; field extensions; counting, approximation and isolating of real roots [1] Chapters 3, 5, 7; [2] Chapters 1, 2, 8, 10; [3] Chapters 1, 2; [6] |
|
| Kirill Shmakov Tarski algorithm with application to elementary geometry and theorem Proving [27]; [28] |
|
| Anton Bankevich Cheneralized Chebyshev polynomials: history and applications of classical Chebyshev polynomials, generation of tree images on complex plane by generalized Chebyshev polynomials [29]; [30]; [31]; [32] |
|
| Daria Romanova Integer Relations among Real Numbers: LLL algorithm, PSLQ algorithm, and applications of integer relations [16]; [17]; [18]; [19]; [20] |
|
| Lukas Bulwahn Computing with Polynomials: Hensel Constructions calculations in Q[x_1, ..., x_n] and Z[x_1, ..., x_n] by inverting morphisms Z[x_1, ..., x_n] -> Z_p[x] via Chinese Remainder Theorem and Hensel lifting, complexity issues [5] Chapters 5, 6; [4] Chapter 15 |
|
| Rosa Freund GCD and factorization of multivariate polynomials [5] Chapters 7, 8 |
|
| Johannes Mittmann Computational problems for polynomial ideals: Hilbert's Nullstellensatz, Dickson's Lemma, term orderings and reductions,Groebner bases, ideal membership, union and intersection of ideals, common zeros and radical ideals, complexity issues, etc. [8]; [4] Chapter 21; [5] Chapter 10; [9] |
|
| Andreas Würfl Basic concepts of differential algebra and applicaiton to indefinite integration and solving ODE's with constant coefficients: differential fields and ideals, field extensions; decidability, connection to Diophantine equations; Risch Algorithm, Louiville Theorem, etc. [5] Chapters 11, 12; [4] Chapters 22, 23; [25] |
|
| Anton Sadovnikov Polynomials and computers: What we cannot do with polynomials [26]; [33]; [34]; [35]; [36] |
|
| Stephan Ritscher Computational problems for differential ideals: infinitness and nonrecursivity of ideals; finitely generated ideals, ideals with finite basis like [x^p], differential orederings and differential reductions; Lee-Brackets, Hall bases for Lee Algebras, Frobenius Theorem and applicaiton to motion controllability [11]; [12]; [13]; [14]; [15] |
|
| Alexey Bogatov Polynomials in Graph Theory: coloring and counting with polynomials [37]; [38]; [39];[40];[41] |
|
| Inna Lukyanenko Cryptography and elliptic curves [22]; [23]; [24]; ... |
Literature: