 English
 فارسی
Computational Algebraic Geometry
If you want to go beyond the linear algebra, then polynomial equations are the next simplest class. Groebner bases are an essential tool for their treatment. The Buchberger algorithm to calculate simultaneously the generalized Gaussian elimination for linear systems of equations and the Euclidean algorithm for finding the greatest common divisor of univariate polynomials. First, Groebner bases are defined for polynomial and its calculation is discussed. Furthermore, it is shown how to many constructive questions of ideal theory to solve algorithmically with their help. Finally, first applications in algebra, linear algebra, number theory, integer programming and algebraic geometry are presented.
The contents:
 Rings, fields, polynomial rings, ideals
 Monomial ideals, Monomial orderings
 Grobner bases and its properties
 Computation of Grobner bases
 Applications of Grobner bases
The references:

David Cox, John Little and Donal O'Shea:
Ideals, varieties, and algorithms
Undergraduate Texts in Mathematics, SpringerVerlag, 2006. 
David Cox, John Little and Donal O'Shea:
Using algebraic geometry
Graduate Texts in Mathematics, SpringerVerlag, 2005. 
Thomas Becker and Volker Weispfenning:
Grobner bases, A computational approach to commutative algebra
Graduate Texts in Mathematics, SpringerVerlag, New York, 1993. 
William W. Adams and Philippe Loustaunau
An introduction to Grobner bases
American Mathematical Society, 1994.
Algebra I,II
TBA
TBA