Some examples for solving systems of algebraic equations by calculating groebner bases

Boege, W.; Gebauer, R.; Kredel, Heinz

doi:10.1016/s0747-7171(86)80014-1

Cited by 81 publications

(26 citation statements)

References 5 publications

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“…It has prime components of degree (2 1 1 12). K6 { The Katsura example, [2], with 6 variables. It has prime components of degree (75 4 1 (2 12 6).…”

Section: Zero Dimensional Triangular Systemsmentioning

confidence: 99%

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Triangular systems and factorized Gröbner bases

Gräbe

1995

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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Solving systems of polynomial equations in an ultimate way means to nd the isolated primes of the associated variety and to present them in a way that is well suited for further computations.[5] proposes an algorithm, that uses several Grobner basis computations for a dimension reduction argument, delaying factorization to the end of the algorithm. In this paper we i n v estigate the opposite approach, heavily using factorization (of multivariate polynomials), delaying the computation of stable ideal quotients. At a heuristic level this is exactly the well known Grobner algorithm with factorization and constraint inequalities, available in all major computer algebra systems. In a preceding paper [9] we reported on some experience with a new version of this algorithm, implemented in our REDUCE package CALI [8]. Here we discuss, how this approach may bere ned to produce triangular systems in the sense of [12] and [13]. Such a re nement guarantees, di erent to the usual Grobner factorizer, to produce a quasi prime decomposition, i.e. the resulting components are at least pure dimensional radical ideals. As in [9] our method weakens the usual restriction to lexicographic term orders.

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“…It has prime components of degree (2 1 1 12). K6 { The Katsura example, [2], with 6 variables. It has prime components of degree (75 4 1 (2 12 6).…”

Section: Zero Dimensional Triangular Systemsmentioning

confidence: 99%

“…E1 -E4 are the examples discussed so far. The remaining examples we took from [2] : G1 and G2 are two variants of Gonnet's example, the original one (G1) and the homogenized as considered in [9] (G2), and H1 is the example Hairer 1. We combined both EFG Band EFG B1with Z S(i.e.…”

Section: Some Examplesmentioning

confidence: 99%

Triangular systems and factorized Gröbner bases

Gräbe

1995

Applied Algebra, Algebraic Algorithms and Error-Correcting Codes

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“…The particular examples that we use come from two articles, one by Boege, Gebauer and Kredel, [2], and the other one by Traverso and Donati, [37]. These examples are listed in Appendix B.…”

Section: Resultsmentioning

confidence: 99%

“…Now, suppose that B and J verify the conditions of the second part of the proposition and that r\^I and r 2 £ L Then (ri) + I fl B ^ 0 and (r 2 ) + I PI B ^ 0. Let r 3 € (n) + P fl5andr 4 e (r 2 ) + P fl P. Then r*3 r 4 € (ri r 2 ) + I 0 B and, therefore, r x r 2 …”

Section: If B Is a Non Empty Multiplicatively Closed Set Of A Commutamentioning

confidence: 99%

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The computation of Gröbner bases on a shared memory multiprocessor

Vidal

1990

Design and Implementation of Symbolic Computation Systems

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Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

Sommese

Verschelde

2000

Journal of Complexity

109

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Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to various polynomial systems, such as the cyclic n-roots problem. Academic PressKey Words: polynomial system; numerical homotopy continuation; components of solutions; numerical algebraic geometry; generic points; embedding. 4. A worked out example. 5. Bertini 's theorem and a local extension theorem. 6. Applications and computational experiences. 6.1. A planar four-bar mechanism. Contents6.2. Constructing Runge-Kutta formulas. 6.3. On Fourier transforms: the cyclic n-roots problem. 7. Conclusions and future directions.

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Some examples for solving systems of algebraic equations by calculating groebner bases

Cited by 81 publications

References 5 publications

Triangular systems and factorized Gröbner bases

Triangular systems and factorized Gröbner bases

The computation of Gröbner bases on a shared memory multiprocessor

Numerical Homotopies to Compute Generic Points on Positive Dimensional Algebraic Sets

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