The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem

The question on the number of points of intersection of curves had been the subject of conjecture for many years. One of the theorems that arise from such conjectures is the Bezout’s theorem. Bezout gives a rigorous proof that when two polynomials in two variables are set equal to 0 simultaneously,...

Full description

Saved in:

Bibliographic Details
Main Author:	Santika, Santika
Format:	Thesis
Language:	English
Published:	2015
Subjects:	QA Mathematics
Online Access:	http://eprints.utm.my/id/eprint/78517/1/SantikaMFS2015.pdf
Tags:	Add Tag No Tags, Be the first to tag this record!

id	my-utm-ep.78517
record_format	uketd_dc
spelling	my-utm-ep.785172018-08-26T11:58:27Z The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem 2015-05 Santika, Santika QA Mathematics The question on the number of points of intersection of curves had been the subject of conjecture for many years. One of the theorems that arise from such conjectures is the Bezout’s theorem. Bezout gives a rigorous proof that when two polynomials in two variables are set equal to 0 simultaneously, one of degree m and the other of degree n, then there cannot be more thanmn solutions unless the two polynomials have a common factor. This is a first form of Bezout’s theorem. In order to have a chance of obtaining a full complement of mn solutions, there have been several adjustments to the first form of Bezout’s theorem, which allow complex solutions instead of just real solutions and considering projective plane curves instead of ordinary plane curves to allow for solutions at infinity. This condition can be realized in the example of two lines, which have no points of intersection on the affine plane if they are parallel. However, in the projective planes, parallel lines do intersect, at a point of the line at infinity. This motivates generalization of the Bezout’s theorem in the number of intersection of projective plane curves under certain conditions. In the case of two variables conics in the affine plane, the resultant can be applied to solve the intersection points. The notion and properties of intersection multiplicity is then applied on each of the intersection points including the points at infinity by considering the homogenization variable. By letting the homogenization variable equals to 1, the properties of intersection multiplicity can be applied to determine the multiplicity of the affine intersection points. By letting the homogenization variable equals to 0, the intersection multiplicity of the points at infinity can also be determined, thus the implementation of the Bezout’s theorem has been illustrated using selected examples. 2015-05 Thesis http://eprints.utm.my/id/eprint/78517/ http://eprints.utm.my/id/eprint/78517/1/SantikaMFS2015.pdf application/pdf en public http://dms.library.utm.my:8080/vital/access/manager/Repository/vital:93809 masters Universiti Teknologi Malaysia, Faculty of Science Faculty of Science
institution	Universiti Teknologi Malaysia
collection	UTM Institutional Repository
language	English
topic	QA Mathematics
spellingShingle	QA Mathematics Santika, Santika The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem
description	The question on the number of points of intersection of curves had been the subject of conjecture for many years. One of the theorems that arise from such conjectures is the Bezout’s theorem. Bezout gives a rigorous proof that when two polynomials in two variables are set equal to 0 simultaneously, one of degree m and the other of degree n, then there cannot be more thanmn solutions unless the two polynomials have a common factor. This is a first form of Bezout’s theorem. In order to have a chance of obtaining a full complement of mn solutions, there have been several adjustments to the first form of Bezout’s theorem, which allow complex solutions instead of just real solutions and considering projective plane curves instead of ordinary plane curves to allow for solutions at infinity. This condition can be realized in the example of two lines, which have no points of intersection on the affine plane if they are parallel. However, in the projective planes, parallel lines do intersect, at a point of the line at infinity. This motivates generalization of the Bezout’s theorem in the number of intersection of projective plane curves under certain conditions. In the case of two variables conics in the affine plane, the resultant can be applied to solve the intersection points. The notion and properties of intersection multiplicity is then applied on each of the intersection points including the points at infinity by considering the homogenization variable. By letting the homogenization variable equals to 1, the properties of intersection multiplicity can be applied to determine the multiplicity of the affine intersection points. By letting the homogenization variable equals to 0, the intersection multiplicity of the points at infinity can also be determined, thus the implementation of the Bezout’s theorem has been illustrated using selected examples.
format	Thesis
qualification_level	Master's degree
author	Santika, Santika
author_facet	Santika, Santika
author_sort	Santika, Santika
title	The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem
title_short	The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem
title_full	The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem
title_fullStr	The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem
title_full_unstemmed	The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem
title_sort	determination of the intersection of projective plane p2 (k) curves based on the bezout's theorem
granting_institution	Universiti Teknologi Malaysia, Faculty of Science
granting_department	Faculty of Science
publishDate	2015
url	http://eprints.utm.my/id/eprint/78517/1/SantikaMFS2015.pdf
_version_	1747818004440154112

The determination of the intersection of projective plane P2 (K) curves based on the bezout's theorem

Similar Items