FGLM algorithm

FGLM is one of the main algorithms in computer algebra, named after its designers, Faugère, Gianni, Lazard and Mora. They introduced their algorithm in 1993. The input of the algorithm is a Gröbner basis of a zero-dimensional ideal in the ring of polynomials over a field with respect to a monomial order and a second monomial order; As its output, it returns a Gröbner basis of the ideal with respect to the second ordering. The algorithm is a fundamental tool in computer algebra and has been implemented in most of the computer algebra systems. The complexity of FGLM is O(nD³), where n is the number of variables of the polynomials and D is the degree of the ideal.^[1] There are several generalization and various application for FGLM.^[2]^[3]^[4]^[5]^[6]

References[]

^ J.C. Faugère; P. Gianni; D. Lazard; T. Mora (1993). "Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering". Journal of Symbolic Computation. 16 (4): 329–344. doi:10.1006/jsco.1993.1051.
^ Middeke, Johannes (2012-01-01). "A Computational View on Normal Forms of Matrices of Ore Polynomials". ACM Commun. Comput. Algebra. 45 (3/4): 190–191. doi:10.1145/2110170.2110182. ISSN 1932-2240.
^ Gerdt, V. P.; Yanovich, D. A. (2003-03-01). "Implementation of the FGLM Algorithm and Finding Roots of Polynomial Involutive Systems". Programming and Computer Software. 29 (2): 72–74. doi:10.1023/A:1022992514981. ISSN 0361-7688.
^ Faugère, Jean-Charles; Mou, Chenqi (2017-05-01). "Sparse FGLM algorithms". Journal of Symbolic Computation. 80, Part 3: 538–569. arXiv:1304.1238. doi:10.1016/j.jsc.2016.07.025.
^ Licciardi, Sandra; Mora, Teo (1994-01-01). Implicitization of Hypersurfaces and Curves by the Primbasissatz and Basis Conversion. Proceedings of the International Symposium on Symbolic and Algebraic Computation. ISSAC '94. New York, NY, USA: ACM. pp. 191–196. doi:10.1145/190347.190416. ISBN 978-0897916387.
^ Borges-Quintana, M.; Borges-Trenard, M. A.; Martínez-Moro, E. (2006-02-20). A General Framework for Applying FGLM Techniques to Linear Codes. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science. Vol. 3857. pp. 76–86. arXiv:math/0509186. doi:10.1007/11617983_7. ISBN 978-3-540-31423-3.

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[1] J.C. Faugère; P. Gianni; D. Lazard; T. Mora (1993). "Efficient Computation of Zero-dimensional Gröbner Bases by Change of Ordering". Journal of Symbolic Computation. 16 (4): 329–344. doi:10.1006/jsco.1993.1051.

[2] Middeke, Johannes (2012-01-01). "A Computational View on Normal Forms of Matrices of Ore Polynomials". ACM Commun. Comput. Algebra. 45 (3/4): 190–191. doi:10.1145/2110170.2110182. ISSN 1932-2240.

[3] Gerdt, V. P.; Yanovich, D. A. (2003-03-01). "Implementation of the FGLM Algorithm and Finding Roots of Polynomial Involutive Systems". Programming and Computer Software. 29 (2): 72–74. doi:10.1023/A:1022992514981. ISSN 0361-7688.

[4] Faugère, Jean-Charles; Mou, Chenqi (2017-05-01). "Sparse FGLM algorithms". Journal of Symbolic Computation. 80, Part 3: 538–569. arXiv:1304.1238. doi:10.1016/j.jsc.2016.07.025.

[5] Licciardi, Sandra; Mora, Teo (1994-01-01). Implicitization of Hypersurfaces and Curves by the Primbasissatz and Basis Conversion. Proceedings of the International Symposium on Symbolic and Algebraic Computation. ISSAC '94. New York, NY, USA: ACM. pp. 191–196. doi:10.1145/190347.190416. ISBN 978-0897916387.

[6] Borges-Quintana, M.; Borges-Trenard, M. A.; Martínez-Moro, E. (2006-02-20). A General Framework for Applying FGLM Techniques to Linear Codes. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Lecture Notes in Computer Science. Vol. 3857. pp. 76–86. arXiv:math/0509186. doi:10.1007/11617983_7. ISBN 978-3-540-31423-3.

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