Theses and Dissertations

Degree

Doctor of Philosophy in Mathematics

Faculty / School

School of Mathematics and Computer Science (SMCS)

Department

Department of Mathematical Sciences

Date of Award

Spring 2026

Advisor

Dr. Junaid Alam Khan, Associate Professor, Department of Mathematical Sciences

Committee Member 1

Dr. Junaid Alam Khan, Supervisor, Institute of Business Administration (IBA), Karachi

Committee Member 2

Dr. Imran Anwar, Examiner-I, Associate Professor, Syed Babar Ali School of Science and Engineering (SBASSE), Lahore University of Management Sciences

Committee Member 3

Dr. Sohail Iqbal, Examiner-II, Assistant Professor, Assistant Professor, COMSAT University Islamabad

Project Type

Dissertation

Access Type

Restricted Access

Pages

ix, 80

Keywords

Gröbner basis, Sagbi basis, Standard basis, Power series, Sasbi basis, strong Gröbner bases, PIR, PID

Abstract

This thesis comprises of three parts: Subalgebra analogue of Standard Bases for Ideals in K[[t1,t2,...,tm]][x1, x2,..., xn], Sasbi-Standard Bases of modules, and Sagbi bases over the product of rings. In the first part of the thesis, we develop a theory for Standard bases of K-subalgebras in K[[t1,t2,...,tm]][x1, x2,..., xn] over a field K with respect to a monomial ordering which is local on t variables and we call them Subalgebra Standard bases. As a second part of the thesis, we evolve a theory for Standard bases of modules over A> (localization of polynomial subalgebra A with respect to >), termed as Sasbi-Standard bases. The main goal is to develop a constructive algorithm for the computation of these bases, which we implement in the computer algebra system SINGULAR. In the last part of the thesis, we give a method to compute (strong) Sagbi bases for subalgebras of a polynomial ring over a commutative ring D = D1 ×D2 × ··· ×Dn provided that these bases are computable in a polynomial ring over commutative ring Di for 1 ≤ i ≤ n.

Download

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

Share

COinS