Presentation of the Book The Gröbner Cover

The aim of this paper is to present the research book “The Gröbner Cover” Montes (The Gröbner Cover, ACM-series, Springer, Berlin, 2019) recently published. This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbn...

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Veröffentlicht in:Mathematics in computer science Jg. 14; H. 2; S. 471 - 482
1. Verfasser: Montes, Antonio
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Cham Springer International Publishing 01.06.2020
Springer Nature B.V
Schlagworte:
Algorithms
Computation
Computer Science
Deduction
Libraries
Loci
Mathematics
Mathematics and Statistics
Polynomials
Taxonomy
Theorems
Geometric locus computation
Geometric envelope computation
08-02 Research exposition
Canonical discussion
13F20 Polynomial rings and ideals
Parametric polynomial systems
Automatic discovery of geometric theorems
Gröbner Cover
11C08 Polynomials
ISSN:1661-8270, 1661-8289
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Abstract The aim of this paper is to present the research book “The Gröbner Cover” Montes (The Gröbner Cover, ACM-series, Springer, Berlin, 2019) recently published. This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbner Cover, to the author’s best knowledge, 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 introduction of Comprehensive Gröbner Systems (CGS), a fundamental contribution made in 1992, and provides a complete description of the Canonical Gröbner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed in Montes, Wibmer (J Symb Comput 45:1391–1425, 2010). 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 hyper-surfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described in the book have also been included in the Singular software library grobcov.lib implemented by the author and H. Schönemann, the book serving also as User Manual for the library.
AbstractList The aim of this paper is to present the research book “The Gröbner Cover” Montes (The Gröbner Cover, ACM-series, Springer, Berlin, 2019) recently published. This book is divided into two parts, one theoretical and one focusing on applications, and offers a complete description of the Canonical Gröbner Cover, to the author’s best knowledge, 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 introduction of Comprehensive Gröbner Systems (CGS), a fundamental contribution made in 1992, and provides a complete description of the Canonical Gröbner Cover (GC), which includes a canonical discussion of a set of parametric polynomial equations developed in Montes, Wibmer (J Symb Comput 45:1391–1425, 2010). 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 hyper-surfaces, and provides algorithms for its computation, as well as for discussing how to determine the real envelope. All the algorithms described in the book have also been included in the Singular software library grobcov.lib implemented by the author and H. Schönemann, the book serving also as User Manual for the library.
Author Montes, Antonio
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Issue 2
Keywords Geometric locus computation
Geometric envelope computation
08-02 Research exposition
Canonical discussion
13F20 Polynomial rings and ideals
Parametric polynomial systems
Automatic discovery of geometric theorems
Gröbner Cover
11C08 Polynomials
Language English
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References MontesAWibmerMGröbner bases for polynomial systems with parametersJ. Symb. Comput.2010451391142510.1016/j.jsc.2010.06.017
Montes, A., Schönemann, H.: “grobcov.lib” A SINGULAR 4-1-2 library for computing the Gröbner Cover of affine rings and applications, (2019)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: SINGULAR 4-1-2-A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, ibinfoyear (2019)
Montes, A.: The Gröbner Cover, ACM-series, Springer, Berlin (2019). ISBN 978-3-030-03904-2
BlazekJPechPLocus computation in dynamic geometry environmentMath. Comput. Sci.201810.1007/s11786-018-0355-307095825
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References_xml – reference: Montes, A.: The Gröbner Cover, ACM-series, Springer, Berlin (2019). ISBN 978-3-030-03904-2
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– reference: BlazekJPechPLocus computation in dynamic geometry environmentMath. Comput. Sci.201810.1007/s11786-018-0355-307095825
– reference: Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: SINGULAR 4-1-2-A computer algebra system for polynomial computations, http://www.singular.uni-kl.de, ibinfoyear (2019)
– reference: Montes, A., Schönemann, H.: “grobcov.lib” A SINGULAR 4-1-2 library for computing the Gröbner Cover of affine rings and applications, (2019)
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SubjectTerms Algorithms
Computation
Computer Science
Deduction
Libraries
Loci
Mathematics
Mathematics and Statistics
Polynomials
Taxonomy
Theorems
Title Presentation of the Book The Gröbner Cover
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