All Theses and Dissertations


Master of Science in Mathematics


Department of Mathematical Sciences

Date of Award

Fall 2015


Dr. Junaid Alam Khan

Committee Member 1

Dr. Muhammad Saeed Akram, COMSATS University, Islamabad

Committee Member 2

Dr. Ahsan Binyamin, Government College University, Faisalabad

Project Type


Access Type

Restricted Access


vii, 30


In this thesis we will define Grobner basis, for ideals of multivariate polynomial ring, which solves the ideal membership problem. Then analogy of the theory of Grobner basis in subalgebra, the theory of Sagbi basis, is discussed. It is shown that Sagbi basis solves the subalgerba membership problem. The main goal is to present the theory related to homogeneous Sagbi basis and d-Sagbi basis. It is shown that homogeneous polynomial with a certain degree d, will, after s-reduction, yield a homogeneous polynomial, having degree within the bound d. Similarly, if a Ssagbi basis is homogeneous then the subalgebra generated by it, is also homogeneous. Lastly it is determined that Sagbi basis construction algorithm bounded by certain degrees, is less expensive for finding Sagbi basis, due to computations within those degree bounds. Moreover it is also presented that bounded algorithm from 0 to certain degree d, gives an outcome, known as d-Sagbi basis. This d-Sagbi basis is used to solve the subalgebra membership problem with a lot less computations as compared to solving it through Sagbi basis.

The full text of this document is only accessible to authorized users.