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Gröbner bases.

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This book demonstrates current trends in research on combinatorial and computational commutative algebra with a primary emphasis on topics related to monomial ideals. Providing a useful and quick introduction to areas of research spanning these fields, Monomial Ideals is split into three parts. Part I offers a quick introduction to the modern theory of Gröbner bases as well as the detailed study of generic initial ideals. Part II supplies Hilbert functions and resolutions and some of the combinatorics related to monomial ideals including the Kruskal—Katona theorem and algebraic aspects of Alexander duality. Part III discusses combinatorial applications of monomial ideals, providing a valuable overview of some of the central trends in algebraic combinatorics. Main subjects include edge ideals of finite graphs, powers of ideals, algebraic shifting theory and an introduction to discrete polymatroids. Theory is complemented by a number of examples and exercises throughout, bringing the reader to a deeper understanding of concepts explored within the text. Self-contained and concise, this book will appeal to a wide range of readers, including PhD students on advanced courses, experienced researchers, and combinatorialists and non-specialists with a basic knowledge of commutative algebra. Since their first meeting in 1985, Juergen Herzog (Universität Duisburg-Essen, Germany) and Takayuki Hibi (Osaka University, Japan), have worked together on a number of research projects, of which recent results are presented in this monograph.

Algebra. --- Mathematics. --- Commutative algebra --- Combinatorial analysis --- Grèobner bases --- Characteristic functions --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Commutative algebra. --- Commutative rings. --- Commutative Rings and Algebras. --- Mathematical analysis --- Characteristic functions. --- Combinatorial analysis. --- Gröbner bases. --- Rings (Algebra)

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This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbner Cover, the most accurate algebraic method for discussing parametric polynomial systems. It also includes applications to the Automatic Deduction of Geometric Theorems, Loci Computation and Envelopes. The theoretical part is a self-contained exposition on the theory of Parametric Gröbner Systems and Bases. It begins with Weispfenning’s introduction of Comprehensive Gröbner Systems (CGS) in 1992, and provides a complete description of the Gröbner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed by Michael Wibmer and the author. In turn, the application part selects three problems for which the Gröbner Cover offers valuable new perspectives. The automatic deduction of geometric theorems (ADGT) becomes fully automatic and straightforward using GC, representing a major improvement on all previous methods. In terms of loci and envelope computation, GC makes it possible to introduce a taxonomy of the components and automatically compute it. The book also generalizes the definition of the envelope of a family of hypersurfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described here have also been included in the software library “grobcov.lib” implemented in Singular by the author, and serve as a User Manual for it.

Gröbner bases. --- Gröbner basis theory --- Commutative algebra --- Algebra. --- Field theory (Physics). --- Algebra --- Commutative Rings and Algebras. --- Field Theory and Polynomials. --- Symbolic and Algebraic Manipulation. --- Data processing. --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis --- Commutative algebra. --- Commutative rings. --- Computer science—Mathematics. --- Rings (Algebra)

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Les équations polynomiales apparaissent dans de nombreux domaines, pour modéliser des contraintes géométriques, des relations entre des grandeurs physiques, ou encore des propriétés satisfaites par certaines inconnues. Cet ouvrage est une introduction aux méthodes algébriques permettant de résoudre ce type d'équations. Nous montrons comment la géométrie des variétés algébriques définies par ces équations, leur dimension, leur degré, ou leurs composantes peuvent se déduire des propriétés des algèbres quotients correspondantes. Nous abordons pour cela des méthodes de la géométrie algébrique effective, telles que les bases de Grobner, la résolution par valeurs et vecteurs propres, les résultants, les bezoutiens, la dualité, les algèbres de Gorenstein et les résidus algébriques. Ces méthodes sont accompagnées d'algorithmes, d'exemples et d'exercices, illustrant leurs applications.

Gröbner bases. --- Polynomials. --- Computational complexity. --- Commutative algebra. --- Algebra --- Complexity, Computational --- Electronic data processing --- Machine theory --- Gröbner basis theory --- Commutative algebra --- Algebra. --- Computer science --- Geometry, algebraic. --- Numerical analysis. --- Commutative Rings and Algebras. --- Computational Mathematics and Numerical Analysis. --- Algebraic Geometry. --- Numerical Analysis. --- General Algebraic Systems. --- Mathematics. --- Mathematical analysis --- Algebraic geometry --- Geometry --- Computer mathematics --- Discrete mathematics --- Mathematics --- Commutative rings. --- Computer mathematics. --- Algebraic geometry. --- Rings (Algebra) --- Grobner bases.

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This edited volume presents a fascinating collection of lecture notes focusing on differential equations from two viewpoints: formal calculus (through the theory of Gröbner bases) and geometry (via quiver theory). Gröbner bases serve as effective models for computation in algebras of various types. Although the theory of Gröbner bases was developed in the second half of the 20th century, many works on computational methods in algebra were published well before the introduction of the modern algebraic language. Since then, new algorithms have been developed and the theory itself has greatly expanded. In comparison, diagrammatic methods in representation theory are relatively new, with the quiver varieties only being introduced – with big impact – in the 1990s. Divided into two parts, the book first discusses the theory of Gröbner bases in their commutative and noncommutative contexts, with a focus on algorithmic aspects and applications of Gröbner bases to analysis on systems of partial differential equations, effective analysis on rings of differential operators, and homological algebra. It then introduces representations of quivers, quiver varieties and their applications to the moduli spaces of meromorphic connections on the complex projective line. While no particular reader background is assumed, the book is intended for graduate students in mathematics, engineering and related fields, as well as researchers and scholars.

Algebra. --- Field theory (Physics). --- Algebraic geometry. --- Associative rings. --- Rings (Algebra). --- Category theory (Mathematics). --- Homological algebra. --- Differential equations. --- Partial differential equations. --- Field Theory and Polynomials. --- Algebraic Geometry. --- Associative Rings and Algebras. --- Category Theory, Homological Algebra. --- Ordinary Differential Equations. --- Partial Differential Equations. --- Gröbner bases. --- Gröbner basis theory --- Commutative algebra --- Partial differential equations --- 517.91 Differential equations --- Differential equations --- Homological algebra --- Algebra, Abstract --- Homology theory --- Category theory (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Topology --- Functor theory --- Algebraic rings --- Ring theory --- Algebraic fields --- Rings (Algebra) --- Algebraic geometry --- Geometry --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Mathematics --- Mathematical analysis