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, Springer-Verlag, 2006.
• David Cox, John Little and Donal O'Shea:
  Using algebraic geometry
  Graduate Texts in Mathematics, Springer-Verlag, 2005.
• Thomas Becker and Volker Weispfenning:
   Grobner bases, A computational approach to commutative algebra
    Graduate Texts in  Mathematics, Springer-Verlag, New York, 1993.
• William W. Adams and Philippe Loustaunau
  An introduction to Grobner bases
  American Mathematical Society, 1994.