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Title: Subalgebra Bases in Local Rings and its Efficient Computation in Polynomial Rings
Keywords: Natural Sciences
Issue Date: 2006
Publisher: GC University Lahore, Pakistan
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 Gr ̈obner walk algorithm and Gr ̈obner 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 as- sumption 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 poly- nomials. In Chapter 4, we study the behavior of Sagbi basis under composition. Some related results are from my paper [21].
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