

Title of Thesis
Subalgebra Bases in Local Rings and its Efficient Computation in
Polynomial Rings 
Author(s)
Junaid Alam Khan 
Institute/University/Department
Details Abdus Salam School of Mathematical Sciences / GC
University, Lahore 
Session 2011 
Subject Mathematics 
Number of Pages 89 
Keywords (Extracted from title, table of contents and
abstract of thesis) Necessary, Subalgebra, Definitions,
Rings, Results, Bases, Polynomials, Computation, Efficient, Local,
Algorithm 
Abstract In Chapter 1, there
are given some necessary definitions and results about monomial
orderings, Standard basis and Sagbi basis in polynomial ring over
the field along with a description on the Gršobner walk algorithm
and Gršobner basis under composition.
In Chapter 2 we develop a theory of subalgebra basis analogous to
Standard basis for ideals in polynomial rings over a field. We call
this basis Sasbi Basis, standing for Subalgebra Analogue to Standard
Basis for Ideals. Sasbi bases may be infinite.In this chapter we
consider subalgebras admitting a finite Sasbi basis and give
algorithms to compute them. Sasbi basis theory is given in my paper
[22].
In Chapter 3, we present an algorithm which converts the Sagbi basis
with respect to one ordering to the Sagbi basis with respect to
another ordering, under the assumption that the subalgebra admits a
finite Sagbi basis with respect to all monomial orderings. We called
it Sagbi walk algorithm.Sagbi Walk algorithm is given in my paper
[20].
Composition is an operation of replacing variables in a polynomial
by other polynomials.In Chapter 4, we study the behavior of Sagbi
basis under composition.Some related results are from my paper [21].

