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The Grobner basis of a module over KUX1,...,Xne and polynomial solutions of a system of linear equations

Authors:

A. Furukawa,

T. Sasaki,

H. KobayashiAuthors Info & Claims

SYMSAC '86: Proceedings of the fifth ACM symposium on Symbolic and algebraic computation

Pages 222 - 224

https://doi.org/10.1145/32439.32483

Published: 01 October 1986 Publication History

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Abstract

Many computations relating polynomial ideals are reduced to calculating polynomial solutions of a system of linear equations with polynomial coefficients[1]. Zacharias[2] pointed out that Buchberger's algorithm[3] for Gröbner basis can be applied to solving such a linear equation. From the computational viewpoint, Zacharias' method seems to be much better than the previous methods. Hence, we have generalized his method to solve a system of equations directly. After completing the paper, we knew that similar works had been done by several authors[4,5]. This paper describes our method briefly.

References

[1]

A, Seidenberg, "Constructions in algebra", Trans. Amer. Math. Soc. 197, pp.273-313, 1974.

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[2]

G. Zacharias, "Generalized GrSbner basis in commutative polynomial rings", Bachelor Thesis, M.I.T., Dept. of Comp. Scie., 1978.

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[3]

B. Buchberger, "Basic features and development of the critical-pair/completion procedure", Prec. First Interntl Conf. on "Rewriting Techniques and Applications", France, 1985, Springer Lecture Notes Comput. Sci.

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[4]

H. M. M~iller and F. Mora, "New constructive methods in classical ideal theory" Journ. Algebra, 99, 1986.

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[5]

D. Bayer, F. Mora, H. M. M611er, private comrnuni cation.

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Yuno TOhtsuka T(2014)Realization of a vector field via state feedback for polynomial dynamical systems2014 European Control Conference (ECC)10.1109/ECC.2014.6862184(2454-2459)Online publication date: Jul-2014
https://doi.org/10.1109/ECC.2014.6862184
Nabeshima K(2010)On the computation of parametric Gröbner bases for modules and syzygiesJapan Journal of Industrial and Applied Mathematics10.1007/s13160-010-0003-z27:2(217-238)Online publication date: 15-May-2010
https://doi.org/10.1007/s13160-010-0003-z
Zhiping Lin Xu LBose N(2008)A Tutorial on GrÖbner Bases With Applications in Signals and SystemsIEEE Transactions on Circuits and Systems I: Regular Papers10.1109/TCSI.2007.91400755:1(445-461)Online publication date: Mar-2008
https://doi.org/10.1109/TCSI.2007.914007
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Index Terms

The Grobner basis of a module over KUX1,...,Xne and polynomial solutions of a system of linear equations
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Computer algebra systems
    2. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices
      2. Computations on polynomials

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SYMSAC '86: Proceedings of the fifth ACM symposium on Symbolic and algebraic computation

October 1986

254 pages

ISBN:0897911997

DOI:10.1145/32439

Editor:
Bruce W. Char
Univ. of Waterloo, Waterloo, Canada

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Published: 01 October 1986

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SYMSAC86: Proceedings of the Symposium on Symbolic and Algebraic Manipulation

July 21 - 23, 1986

Ontario, Waterloo, Canada

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Yuno TOhtsuka T(2014)Realization of a vector field via state feedback for polynomial dynamical systems2014 European Control Conference (ECC)10.1109/ECC.2014.6862184(2454-2459)Online publication date: Jul-2014
https://doi.org/10.1109/ECC.2014.6862184
Nabeshima K(2010)On the computation of parametric Gröbner bases for modules and syzygiesJapan Journal of Industrial and Applied Mathematics10.1007/s13160-010-0003-z27:2(217-238)Online publication date: 15-May-2010
https://doi.org/10.1007/s13160-010-0003-z
Zhiping Lin Xu LBose N(2008)A Tutorial on GrÖbner Bases With Applications in Signals and SystemsIEEE Transactions on Circuits and Systems I: Regular Papers10.1109/TCSI.2007.91400755:1(445-461)Online publication date: Mar-2008
https://doi.org/10.1109/TCSI.2007.914007
Fujisaka HSato C(2007)An algebraic analysis for bifurcation problemsElectronics and Communications in Japan (Part III: Fundamental Electronic Science)10.1002/ecjc.443074090374:9(22-32)Online publication date: 21-Feb-2007
https://doi.org/10.1002/ecjc.4430740903
Sakata S(2005)Finding a minimal polynomial vector set of a vector of nD arraysApplied Algebra, Algebraic Algorithms and Error-Correcting Codes10.1007/3-540-54522-0_129(414-425)Online publication date: 7-Jun-2005
https://doi.org/10.1007/3-540-54522-0_129
Weispfenning V(2005)Gröbner bases for polynomial ideals over commutative regular ringsEurocal '8710.1007/3-540-51517-8_137(336-347)Online publication date: 31-May-2005
https://doi.org/10.1007/3-540-51517-8_137
Lin ZXu LWu Q(2004)Applications of Gröbner bases to signal and image processing: a surveyLinear Algebra and its Applications10.1016/j.laa.2004.01.008391(169-202)Online publication date: Dec-2004
https://doi.org/10.1016/j.laa.2004.01.008
Mori K(2003)A new parameterization method for all stabilizing controllers of nD systems without coprime factorizabilityProceedings of the 2003 International Symposium on Circuits and Systems, 2003. ISCAS '03.10.1109/ISCAS.2003.1205110(III-678-III-681)Online publication date: 2003
https://doi.org/10.1109/ISCAS.2003.1205110
Xu LSaito OAbe K(1994)Output feedback stabilizability and stabilization algorithms for 2D systemsMultidimensional Systems and Signal Processing10.1007/BF009858625:1(41-60)Online publication date: 1-Jan-1994
https://dl.acm.org/doi/10.1007/BF00985862
Möller H(1991)Computing Syzygies à la Gauß-JordanEffective Methods in Algebraic Geometry10.1007/978-1-4612-0441-1_22(335-345)Online publication date: 1991
https://doi.org/10.1007/978-1-4612-0441-1_22
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Recurrence equations and their classical orthogonal polynomial solutions

Computing a Gröbner basis of a polynomial ideal over a Euclidean domain

Chebyshev polynomial solutions of systems of linear integral equations

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Recurrence equations and their classical orthogonal polynomial solutions

Computing a Gröbner basis of a polynomial ideal over a Euclidean domain

Chebyshev polynomial solutions of systems of linear integral equations

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