Bruno Buchberger

main contributions
refereed publications
books - journals
invited talks
contributed talks

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).

By now, five text books and more than 300 journal and conference articles have been published worldwide on the theory of Groebner bases. Over the past ten years, my papers on Groebner bases have been cited over 1000 times in refereed journals (see the CompuMath Citation Index.) The American Mathematical Society key word index for mathematics has recently created an extra key word "Groebner bases". The Groebner basis algorithm is now contained in all major computer algebra software systems and is installed in several million copies of these systems worldwide.