Preface
This bibliography was initiated by Bruno Buchberger and is being built up under his direction. The authors of the bibliography are Bruno Buchberger and Alexander Zapletal and the managing editors are Buchberger's PhD students M.Wiesinger and H.Rahkooy.
In this bibliography, we try to collect all journal and proceeding articles, books, survey articles, technical reports, tutorials, lecture notes, slides of talks etc. on Gröbner bases theory and related topics (theory, algorithms, systems, applications).
As far as this is possible and reasonable, items in the bibliography are downloadable or links to downloadables are provided.
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Introduction of the Algorithmic Theory of Gröbner Bases
- An Algorithm for Finding the Basis Elements in the Residue Class Ring Modulo a Zero Dimensional Polynomial Ideal.
B. Buchberger
PhD Thesis, Mathematical Institute, University of Innsbruck, Austria, 1965.
German version:
Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal.
- An Algorithmic Criterion for the Solvability of Algebraic Systems of Equations.
B. Buchberger
Aequationes mathematicae 4/3, 1970, pp. 374-383. (English translation in: B. Buchberger, F. Winkler (eds.), Gröbner Bases and Applications, Proceedings of the International Conference "33 Years of Gröebner Bases", 1998, RISC, Austria, London Mathematical Society Lecture Note Series, Vol. 251, Cambridge University Press, 1998, pp. 535 -545.)
German version:
Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems.
Early Forerunners
- Euclid
Non-linear univariate systems. Euclid's algorithm turns out to be a special case of the Gröbner bases algorithm.
- C.F. Gauss
Linear multivariate systems. Gauss's algorithm turns out to be a special case of the Gröbner bases algorithm.
- Division Process in 19th Century
A kind of division process for multivariate polynomials.
- P. Gordan, 1899
Introduced (in the language of that time) the notion of Gröbner bases (based on the division process) and showed their existence (finiteness?) but did not give an algorithm for constructing them.
- L.E. Dickson, 1913
A general lemma on the termination of certain sequences of tuples of natural numbers.
From this lemma, the termination of the Gröbner bases algorithm follows easily. And, then, Hilbert's basis theorem is an easy corollary of the existence of Gröbner bases.
- M. Janet, 1920
Similar notions and methods for differential equations (viewed symbolically).
- G. Hermann, 1926
A solution of the membership problem for polynomial ideals based on linear algebra.
- W. Gröbner
- J. Ritt
Characteristic sets.
- A.I. Shirshov, 1962
Similar notion and method for Lie algebras.
- H. Hironaka, 1964
In the frame of his famous paper on the resolution of singularities: the notion of "standard bases" in the domain of formal power series, but no algorithm for their construction.
Textbooks and Surveys
- A Singular Introduction to Commutative Algebra (Gert-Martin Greuel, Gerhard Pfister)
- Algorithms for Computer Algebra (K. Geddes, S.R. Czapor, G. Labahn)
- An Introduction to Gröbner Bases (W. Adams, P. Loustaunau)
- An Introduction to Gröbner Bases (R. Froeberg)
- Commutative Algebra (D. Eisenbud)
- Computational Commutative Algebra 1 (M. Kreuzer, L. Robbiano)
- Computational Fundamentals of Gröbner Bases (M. Noro, K. Yokoyama)
- Gröbner Bases and Applications (B. Buchberger, F. Winkler)
- Gröbner Bases and Their Applications (M. Masaki)
- Gröbner Bases (T. Becker, V. Weispfenning, Heinz Kredel)
- Gröbner deformations of hypergeometric differential equations (B. Sturmfels, M. Saito, N.Takayama)
- Ideals, Varieties, and Algorithms (D. Cox, J. Little, D. O'Shea)
- Modern Computer Algebra (J. von zur Gathen, J. Gerhard)
- Polynomial Algorithms in Computer Algebra (F. Winkler)
- Using Algebraic Geometry (D. Cox, J. Little, D. O'Shea)
- Introduction to Gröbner Bases (B. Buchberger)
-
Gröbner-Bases: An Algorithmic Method in Polynomial Ideal Theory. (B. Buchberger)
Chapter 6 in: N.K. Bose (ed.), Multidimensional Systems Theory - Progress, Directions and Open Problems in Multidimensional Systems, Reidel Publishing Company, Dodrecht - Boston - Lancaster, 1985, pp. 184-232. (Second edition: N.K.Bose (ed.): Multidimensional Systems Theory and Application, Kluwer Academic Publisher, 2003, pp.89-128.)
- Solving polynomial equation systems. I: The Kronecker-Duval philosophy (F. Mora)
- Solving Polynomial Equation Systems II : Macaulay's Paradigm and Gröbner Technology (F. Mora)
- Noncommutative Gröbner Bases and Filtered-Graded Transfer (H. Li)
- Algorithmic Methods in Non-Commutative Algebra : Applications to Quantum Groups (J. Bueso, J. Gomez Torrecillas, A. Verschoren)
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