### 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)

--- the 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.

#### The Theorema Project#

The Theorema project is also a part of the SFB (Special Research Consortium) "Scientific Computing" at the University of Linz, sponsored by the FWF (Austrian National Science Foundation).

The Theorema Project is my main research interest since 1995. However, basically, all my previous math activities can be view as preparatory work to the Theorema Project: In an oversimplified view, the objective of the Theorema project is the development of a software system that simulates my own proof methods and proof style which I apply in research and I also teach as a part of my "Thinking, Speaking, Writing" course. More generally, the Theorema software system is a uniform logic and software technologic frame for supporting all phases of the mathematical exploration cycle: formalization, proving, solving, computing. The system is programmed in Mathematica and consists of the following parts:

--- a syntax analyzer that accepts two-dimensional mathematical formulae in a notation that is very close to the "usual" mathematical notation in textbooks

--- a mechanism for expressing functors

--- a formal language that allows to build up hierarchies of labeled formulae in structured mathematical knowledge bases

--- various general and special provers for the automated generation of proofs for various classes of mathematical formulae (e.g. general predicate logic formulae, equalities over natural numbers and other inductive domains, boolean combinations of equalities over the complex numbers, set theory formulae, etc.)

--- post-processors that present the proofs generated in various natural languages (at the moment English and Japanese),
interfaces for sending formulae and knowledge bases from Theorema to various external provers like Otter etc. and for translating the output of these provers back to Theorema syntax

--- a mechanism for accessing all Mathematica functions from within Theorema

Recently I introduced the notion of "logicographic" symbols that open new possibilities for combining formal reasoning with graphical intuition.

#### Decomposition of Goedel Numberings#

Operational semantics of programming languages and the study of Goedel numberings was my main research interest in the period 1968-1976. I wanted to find necessary and sufficient conditions for making a computability mechanism universal. My main contribution was the notion of a recursive automaton decomposition of Goedel numberings and the proof that every Goedel numbering (abstract model of a universal programming language) can be decomposed into recursive input / transition / and output functions in a natural way. I finally also managed to characterize the possible input / transition / and output functions of Goedel numberings (universal programming languages).

#### Computer-Trees and the L-Machine#

From 1976 until 1985, my main research interest was the parallelization of symbolic computation algorithms and the construction of appropriate parallel machines. I introduced the notion of "Computer Trees" and, later, the notion of "L-machine" as a computational model and as a basis for an early distributed memory parallel machine. In 1978-1983, I built two versions of the L-machine that consisted of 8 processors in cooperation with an Austrian engineering company. I was able to implement various experimental parallel algorithms on this early parallel machine for demonstrating the future effects of parallelization.

#### P-adic Arithmetic: #

In 1986, I did some detailed analysis on P-adic arithmetics as an alternative in computer algebra systems for speeding up these systems.

#### Hybrid (Symbolic and Neural Network Based) Approach to Robotics: #

In the frame of the Japanese "Real World Computing" project, 1993 and later, W. Jacak and myself co-directed a project on hybrid (symbolic / numeric / neural networks) methods for robotics. In the frame of this project, we developed algorithms that produce the neural network control for robots from a symbolic description of the robot kinematics.

#### Systolic Algorithms for Computer Algebra:#

In 1993 and later, I was involved in the research mainly pursued by T. Jebelean on speeding up rational arithmetic by systolic areas as a basis for speeding up algorithms in the area of polynomial ideal theory

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