Sulaiman, Surajo (2018) New methods of computing the projective polynomial resultant based on dixon, jouanolou and jacobian matrices. EngD thesis, Universiti Teknologi Malaysia.

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Abstract
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 JouanolouJacobian method, the DixonJouanolou methods for bivariate polynomials, and their generalizations to the multivariate case. The DixonJouanolou 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 DixonJouanolou method for the bivariate systems of three polynomials to systems of n+1 polynomials with n variables. In order to implement the DixonJouanolou 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, pseudohomogenization 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 DixonJouanolou construction and implementation, it is found that the DixonJouanolou 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 multipolynomial divisions. The JouanolouJacobian 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.
Item Type:  Thesis (EngD) 

Uncontrolled Keywords:  quartic, JouanolouJacobian method, bivariate systems 
Subjects:  Q Science > Q Science (General) 
Divisions:  Science 
ID Code:  81395 
Deposited By:  Narimah Nawil 
Deposited On:  23 Aug 2019 04:07 
Last Modified:  23 Aug 2019 04:07 
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