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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].

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S. No. Chapter Title of the Chapters Page Size (KB)
1 0 CONTENTS

 

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1

PRELIMINARIES

1.1 Monomial Ordering
1.2 Standard Bases
1.3 Sagbi Basis
1.4 Gršobner Walk
1.5 Gršobner basis under Composition

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3 2 SUBALGEBRA ANALOGUE TO STANDARD BASIS FOR IDEAL

2.1 Sasbi basis of F[[Γ]]
2.2 Sasbi Basis in the Localization of F[x1; : : : ; xn]
2.3 Implementation in SINGULAR

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4 3 CONVERTING SUBALGEBRA BASES WITH THE SAGBI WALK

3.1 Sagbi cone
3.2 Sagbi Walk
3.3 Implemetation in SINGULAR

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5 4 SAGBI BASIS UNDER COMPOSITION

4.1 Main Results
4.2 Examples
4.3 Implementation in SINGULAR

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BIBLIOGRAPHY

 

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