Comprehensive Gröbner bases

Weispfenning, Volker

doi:10.1016/0747-7171(92)90023-w

Cited by 229 publications

(127 citation statements)

References 11 publications

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“…This conjecture can be rephrased in terms of universal comprehensive Gröbner bases [24]: the parametric ideal I 2 (L) (the parameters being the a ijk 's) has a comprehensive and universal Gröbner basis whose elements are multihomogeneous of degree bounded by (1, 1, . .…”

Section: Introductionmentioning

confidence: 99%

Linear spaces, transversal polymatroids and ASL domains

Conca

2006

J Algebr Comb

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Section: Introductionmentioning

confidence: 99%

Linear spaces, transversal polymatroids and ASL domains

Conca

2006

J Algebr Comb

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“…The study of parametric polynomial systems via Gröbner bases was initiated by [38]. Since then, many improvements to this theory have been made (e.g.…”

Section: Using Gröbner Systems To Detect Degenerate Componentsmentioning

confidence: 99%

Automatic deduction in (dynamic) geometry: Loci computation

Botana

Abánades

2014

Computational Geometry

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A symbolic tool based on open source software that provides robust algebraic methods to handle automatic deduction tasks for a dynamic geometry construction is presented. The prototype has been developed as two different worksheets for the open source computer algebra system Sage, corresponding to two different ways of coding a geometric construction, namely with the open source dynamic geometry system GeoGebra or using the common file format for dynamic geometry developed by the Intergeo project. Locus computation algorithms based on Automatic Deduction techniques are recalled and presented as basic for an efficient treatment of advanced methods in dynamic geometry. Moreover, an algorithm to eliminate extraneous parts in symbolically computed loci is discussed. The algorithm, based on a recent work on the Gröbner cover of parametric systems, identifies degenerate components and extraneous adherence points in loci, both natural byproducts of general polynomial algebraic methods. Several examples are shown in detail.

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“…It may be useful to look at Theorem 1 in the context of what are known as comprehensive Gröbner bases [2] [17] [24]. This construction in effect allows one to circumvent the problem that Gröbner bases are not continuous in their input data; indeed it seems designed to address that defect.…”

Section: Proof (Sketch)mentioning

confidence: 99%

Cylinders Through Five Points: Complex and Real Enumerative Geometry

Lichtblau

2007

Automated Deduction in Geometry

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Abstract. It is known that five points in 3 generically determine a finite number of cylinders containing those points. We discuss ways in which it can be shown that the generic (complex) number of solutions, with multiplicity, is six, of which an even number will be real valued and hence correspond to actual cylinders in 3 . We partially classify the case of no real solutions in terms of the geometry of the five given points. We also investigate the special case where the five given points are coplanar, as it differs from the generic case for both complex and real valued solution cardinalities.

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Comprehensive Gröbner bases

Cited by 229 publications

References 11 publications

Linear spaces, transversal polymatroids and ASL domains

Linear spaces, transversal polymatroids and ASL domains

Automatic deduction in (dynamic) geometry: Loci computation

Cylinders Through Five Points: Complex and Real Enumerative Geometry

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