Gröbner bases have become a key tool in many elimination techniques and in computer calculations. They were introduced in the thesis of B. Buchberger, who was one of two people responsible for a special semester devoted to Gröbner bases and related methods at the Radon Institute of Computational and Applied Mathematics in Linz. This book brings together 11 survey papers based on lectures presented in workshops organized during the semester and they cover a broad territory. Use of Gröbner bases in algebraic analysis (including applications to control theory) is treated in the paper by J.-F. Pommaret. Applications of Gröbner bases to linear partial differential and difference equations are described by U. Oberst and F. Pauer and applications to computations of the strength of systems of differential equations are covered in the paper by A. Levin. Differential Gröbner bases are discussed in the contribution by G. Carrà Ferro and differential elimination and its applications in biological modelling are described by F. Boulier. The Janet algorithm, involutive bases and generalized Hilbert series are treated by D. Robertz. The paper by W. M. Seiler is devoted to the Spencer cohomology, homological formulation of the Cartan test for involutivity of a system and its dual version, involutive bases (in particular the Pommaret bases) and applications to general differential equations. Local Lie groups of transformations have an infinite-dimensional analogue called Lie pseudo-groups. Their description can be found in the paper by P. Olver and J. Pohjanpelto, including a generalization of the method of moving frames developed first in a finite dimensional situation. They also describe algorithms for the computation of algebra of differential invariants. Constructive techniques in invariant theory for rings of differential operators are studied in the contribution by W. N. Traves (including a computation of the ring of invariants for the case of the Grassmannian G2,4). The paper by K. Krupchyk and J. Tuomela discusses compatibility complexes for overdetermined boundary problems. The last contribution (F. L. Pritchard and W. Y. Sit) treats initial value problems for ordinary differential-algebraic equations.

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