
Theory
of "Groebner Bases'' / The
Theorema Project / Decomposition
of Goedel Numberings / ComputerTrees
and the LMachine / Padic
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 "Spolynomials'' (1965),
 the main theorem about Groebner bases and Spolynomials 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 nonlinear 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.

