New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices

In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less...

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Main Author:	Sulaiman, Surajo
Format:	Thesis
Language:	English
Published:	2018
Subjects:	Q Science (General)
Online Access:	http://eprints.utm.my/id/eprint/81395/1/SurajoSulaimanPFS2018.pdf
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spelling	my-utm-ep.813952019-08-23T04:07:04Z New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices 2018 Sulaiman, Surajo Q Science (General) In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less superfluous factors. In this thesis, three different techniques for computing the resultant matrix are presented, namely the Jouanolou-Jacobian method, the Dixon-Jouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The Dixon-Jouanolou method is proposed based on the existing Jouanolou matrix method which is subjected to bivariate systems. To further extend this method to multivariate systems, the entry formula for computing the Dixon resultant matrix is first generalized. This extended application of the loose entry formula leads to the possibility of generalizing the Dixon-Jouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the Dixon-Jouanolou method on systems of polynomials over the affine and projective space, respectively, the concept of pseudohomogenization is introduced. Each space is subjected to its respective conditions; thus, pseudo-homogenization serves as a bridge between them by introducing an artificial variable. From the computing time analysis of the generalized loose entry formula used in the computation of the Dixon matrix entries, it is shown that the method of computing the Dixon matrix using this approach is efficient even without the application of parallel computations. These results show that the cost of computing the Dixon matrix can be reduced based on the number of additions and multiplications involved when applying the loose entry formula. These improvements can be more pronounced when parallel computations are applied. Further analyzing the results of the hybrid Dixon-Jouanolou construction and implementation, it is found that the Dixon-Jouanolou method had performed with less computational cost with cubic running time in comparison with the running time of the standard Dixon method which is quartic. Another independent construction produced in this thesis is the Jouanolou- Jacobian method which is an improvement of the existing Jacobian method since it avoids multi-polynomial divisions. The Jouanolou-Jacobian method is also able to produce a considerably smaller resultant matrix compared to the existing Jacobian method and is therefore less computationally expensive. Lastly all the proposed methods have considered a systematic way of detecting and removing extraneous factors during the computation of the resultant matrix whose determinant gives the polynomial resultant. 2018 Thesis http://eprints.utm.my/id/eprint/81395/ http://eprints.utm.my/id/eprint/81395/1/SurajoSulaimanPFS2018.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:125012 engd doctoral Universiti Teknologi Malaysia Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	Q Science (General)
spellingShingle	Q Science (General) Sulaiman, Surajo New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
description	In elimination theory, particularly when using the matrix method to compute multivariate resultant, the ultimate goal is to derive or construct techniques that give a resultant matrix that is of considerable size with simple entries. At the same time, the method should be able to produce no or less superfluous factors. In this thesis, three different techniques for computing the resultant matrix are presented, namely the Jouanolou-Jacobian method, the Dixon-Jouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The Dixon-Jouanolou method is proposed based on the existing Jouanolou matrix method which is subjected to bivariate systems. To further extend this method to multivariate systems, the entry formula for computing the Dixon resultant matrix is first generalized. This extended application of the loose entry formula leads to the possibility of generalizing the Dixon-Jouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the Dixon-Jouanolou method on systems of polynomials over the affine and projective space, respectively, the concept of pseudohomogenization is introduced. Each space is subjected to its respective conditions; thus, pseudo-homogenization serves as a bridge between them by introducing an artificial variable. From the computing time analysis of the generalized loose entry formula used in the computation of the Dixon matrix entries, it is shown that the method of computing the Dixon matrix using this approach is efficient even without the application of parallel computations. These results show that the cost of computing the Dixon matrix can be reduced based on the number of additions and multiplications involved when applying the loose entry formula. These improvements can be more pronounced when parallel computations are applied. Further analyzing the results of the hybrid Dixon-Jouanolou construction and implementation, it is found that the Dixon-Jouanolou method had performed with less computational cost with cubic running time in comparison with the running time of the standard Dixon method which is quartic. Another independent construction produced in this thesis is the Jouanolou- Jacobian method which is an improvement of the existing Jacobian method since it avoids multi-polynomial divisions. The Jouanolou-Jacobian method is also able to produce a considerably smaller resultant matrix compared to the existing Jacobian method and is therefore less computationally expensive. Lastly all the proposed methods have considered a systematic way of detecting and removing extraneous factors during the computation of the resultant matrix whose determinant gives the polynomial resultant.
format	Thesis
qualification_name	engd
qualification_level	Doctorate
author	Sulaiman, Surajo
author_facet	Sulaiman, Surajo
author_sort	Sulaiman, Surajo
title	New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
title_short	New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
title_full	New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
title_fullStr	New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
title_full_unstemmed	New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
title_sort	new methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices
granting_institution	Universiti Teknologi Malaysia
granting_department	Science
publishDate	2018
url	http://eprints.utm.my/id/eprint/81395/1/SurajoSulaimanPFS2018.pdf
_version_	1747818321420484608

New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices

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