Theory of "Groebner Bases'' / The Theorema Project / Decomposition of Goedel Numberings / Computer-Trees and the L-Machine / P-adic Arithmetic / Hybrid Approach to Robotics / Systolic Algorithms for Computer Algebra

Main Contributions / Theory of Groebner Bases

In my PhD thesis 1965 I initiated the theory of "Groebner bases" by which quite a few fundamental problems in algebraic geometry (commutative algebra) can be solved algorithmically. In various periods of my life I turned back to the development of this theory. My main achievements in the theory of Groebner bases are:

• the notion of "Groebner Bases''(1965),
  
• he notion of "S-polynomials'' (1965),
  
• the main theorem about Groebner bases and S-polynomials on which an algorithm for constructing Groebner bases is based(1965),
  
• the notion of "reduced Groebner basis'' (1965),
  
• a general termination proof (1970),
  
• a first computer implementation of the algorithm (1965)
  
• first computational examples (1965),
  
• first applications in the area of polynomial ideals: computation in residue class rings, Hilbert functions(1965),
  
• solution of algebraic systems (1970),
  
• bases transformations (1970),
  
• improved versions of the Groebner basis algorithm based on the notion of "criteria'' (1979),
  
• a first complexity analysis (1965),
  
• various applications in non-linear geometry (1989),
  
• generalization of the theory of Groebner bases to "reduction rings'' (1983),
  
• generalizations of the criteria to rewrite systems (1983),
  
• improved versions of the proof of the main theorem (1976, 1983).