All Theses and Dissertations


Doctor of Philosophy in Mathematics

Faculty / School

School of Mathematics and Computer Science (SMCS)


Department of Mathematical Sciences

Date of Award

Fall 2021


Junaid Alam Khan

Committee Member 1

Junaid Alam Khan, Associate Professor, School of Mathematics and Computer Science (SMCS), Institute of Business Administration, Karachi

Committee Member 2

Dr. Muhammad Imran Qureshi, COMSATS University, Islamabad

Committee Member 3

Dr. Shamsa Kanwal, Higher Education Department, Govt. of Punjab

Project Type


Access Type

Restricted Access


xv, 66


There are three parts in this thesis, which generally deal with standard bases for modules over polynomial subalgebras, commutation of Sagbi-Gröbner bases and Homogeneous Sagbi bases under the composition of polynomials. These are described in greater detail as follows. In the first part, we present the theory of standard bases for the submodules of free modules of subalgebras over polynomial subalgebras (with finitely many indeterminates). These are based on arbitrary gradings (induced by an additive monoid Γ) which we call "SΓ-bases" (stand for Γ-bases over subalgebras). Here, we develop theory, and algorithms, for the construction and computation of these bases. In the second part, we discuss the behaviour of Sagbi-Gröbner bases (bases for ideals of subalgebras of polynomial). This behaviour is discussed under the composition of polynomials with as many variables as in the polynomial rings over a field. We develop sufficient and necessary condition under which the composition by Θ commute with Sagbi-Gröbner bases computation. In the last part, we deal with the problem of the commutation of Homogeneous Sagbi bases with polynomial composition. Moreover, we present a sufficient and necessary criterion on a set Θ to ensure that for any Homogeneous Sagbi basis S, the composed set S ◦Θ is again a Homogeneous Sagbi basis.

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