**Course No.****：**S070104J06

**Course Category****：**Professional Basic Course

**Period/Credits****：**40/2

**Prerequisites****：**Advanced Algebra, Modern Algebra.

**Aims & Requirements****：**

The course are designed for the graduate students in basic mathematics, applied mathematics, computational mathematics and computer science. It is suitable to take as an elective course for the students interesting in cryptography, information science, system science. The course will focus on the classical theories and algorithms of computational algebra, as well as symbolic computation of calculus. All the knowledge is provided to a good foundation for students studying relative subjects.

**Primary Coverage****：**

Chapter 1 Introduction

Computer algebra and mathematical computation, Computer algebra system, Problems and demonstrations.

Chapter 2 Data representation and foundational computation

Big integral number computation, Polynomials representation and symbolic computation, Congruence and interpolation.

Chapter 3 Resultant and subresultant

Unary resultant, Common zeros of polynomials and multiply roots criteria, Subresultant chain theorem, Dixon and Macaulay resultants, Application of resultant.

Chapter 4 Modulo computation of polynomials

Polynomial division with remainder, Subresultant and remainder sequence, Chinese remainder theorem, to compute the great common divisors, P-adic representation, Newton iteration, decomposition of free square factors, decomposition of unary polynomials over finite field, decomposition of multi polynomials over finite field.

Chapter 5 Characteristic sequences

Triangular sequence and characteristic sequence, Wu-Ritt algorithm, Irreducible triangular sequence, Regular triangular sequence, decomposition of zeros of polynomials, Proving geometric theorems.

Chapter 6 Greobner bases

Term order, Reduction of polynomials, Greobner basis, Buchberger algorithms, Computational ideal theory, Solving systems of polynomial equations.

**Textbook****：**

Chen Yufu, Lectures on computer algebra, Higher Education Press, 2009.

**References****：**

[1] Wang Dongming, Xia Bichan and Li Ziming, Computer Algebra, Qinghua University Press, 2007.

[2] J. von zur Gathen and J.Gerhard, Modern Computer Algebra，Combridge University press, 1999.

[3] K.O.Geddes, S.R.Czapor and G.Labahn , Algorithm for Computer Algebra, Kluwer Academic Publishers, Sixth Printing, 1999.

[4] B.Mishra, Algorithmic Algebra, Springer-Verlag, 2001.

[5] Wu Wen-tsun , Mathematics Mechanization，Science Press/Kluwer，Beijing 2000.

**Author****：**Yufu Chen (School of Mathematical Sciences, GUCAS)

**Date****：**August, 2009