On the Finding of Final Polynomials

Bokowski, Jürgen; Richter, Jürgen

doi:10.1016/s0195-6698(13)80052-2

Cited by 28 publications

(42 citation statements)

References 9 publications

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“…Bi-quadratic final polynomials as introduced by J. Bokowski and J. RichterGebert [2], [26] are a method to prove non-realizability for oriented matroids [1]. bi-quadratic final polynomials can be considered as a specialization of the more general structure of final polynomials as introduced by J. Bokowski and B. Sturmfels [7].…”

Section: Bi-quadratic Expressionsmentioning

confidence: 99%

“…For this we linearize the problem in the same spirit as it was done in [2], where we the oriented matroid case was considered. To make the algorithmic and algebraic background more transparent, we are going to study the structure detached from the concrete application to our bracket calculations.…”

Section: Translate Any Comparable Pair Of Brackets ([G][c]) Where Gmentioning

confidence: 99%

“…The prover we want to present in this chapter works also on the bracket algebra level and makes use of the bi-quadratic final polynomial method introduced in [2], [26]. Here any incidence relation is represented by a whole class of biquadratic equations.…”

Section: Introductionmentioning

confidence: 99%

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Mechanical theorem proving in projective geometry

Richter-Gebert

1995

Ann Math Artif Intell

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Section: Bi-quadratic Expressionsmentioning

confidence: 99%

Section: Translate Any Comparable Pair Of Brackets ([G][c]) Where Gmentioning

confidence: 99%

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Mechanical theorem proving in projective geometry

Richter-Gebert

1995

Ann Math Artif Intell

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“…The prover proposed by Richter-Gebert (1995) is based on the final biquadratic polynomial method (see also Bokowski and Richter-Gebert, 1990;Sturmfels, 1989). A proof produced by this prover is extremely short and geometrically meaningful.…”

Section: Introductionmentioning

confidence: 99%

Automated Theorem Proving in Incidence Geometry — A Bracket Algebra Based Elimination Method

2001

Automated Deduction in Geometry

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Abstract.In this paper we propose a bracket algebra based elimination method for automated generation of readable proofs for theorems in incidence geometry. This method is based on two techniques, the first being some heuristic elimination rules which improve the performance of the area method of without introducing signed length ratios, the second being a simplification technique called contraction, which reduces the size of bracket polynomials. More than twenty theorems in incidence geometry have been proved, for which short proofs are produced swiftly. An interesting phenomenon is that a proof composed of polynomials of at most two terms can always be found for any of these theorems, similar to that by the final biquadratic polynomial method of Richter-Gebert (1995).

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“…recursively add new facets to the list (9) evaluate whenever there are enough facets in the list (10) facet list = facet list \ {F} After roughly two weeks of computation on standard Linux workstations with altogether 45 kernels, the algorithm had enumerated all connected 2s2s rank-5 Eulerian lattices with up to 12 vertices. This produced exactly the face lattices of the spheres listed in Theorem 2.1, and thus proves that theorem as well as the second part of Theorem 1.1.…”

mentioning

confidence: 99%

A Flag Vector of a 3‐sphere That Is Not the Flag Vector of a 4‐polytope

Brinkmann¹,

Ziegler²

2016

Mathematika

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We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters (f 0 , f 1 , f 2 , f 3 ; f 02 ) = (12, 40, 40, 12; 120), but this sphere is not realizable by a convex 4-polytope.The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

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On the Finding of Final Polynomials

Cited by 28 publications

References 9 publications

Mechanical theorem proving in projective geometry

Mechanical theorem proving in projective geometry

Automated Theorem Proving in Incidence Geometry — A Bracket Algebra Based Elimination Method

A Flag Vector of a 3‐sphere That Is Not the Flag Vector of a 4‐polytope

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