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Ziada, Samira (Beograd , 2015)[more][less]
Abstract: This thesis has been written under the supervision of my mentor Prof. Aleksandar T. Lipkovski at the University of Belgrade in the academic year 2014- 2015. The topic of this thesis is Classi cation of Monomial Orders In Polynomial Rings and Gr obner Basis. The thesis is divided into four chapters let us review the content and the main contributions to this doctoral thesis. In the rst chapter (page. 1-6), the de nitions, features and examples of the notions basic for this thesis are given. Also, relevant properties of the multivariate polynomial ring and the division with reminder algorithm are recalled. In Chapter 2 (page. 7-22), the classi cation of monomial orderings for multivariate polynomial rings is displayed (given) in detail, with special emphasis on the case of two variables. The connection of this classi cation and the well known classi cation of Robiano is exposed in detail. It is an unusual and a little-known fact that the set of di erent monomial orderings with the natural topology is a Cantor set. Chapter 3 (page. 23-40) and Chapter 4 (page. 41-51) contain the main contributions of the thesis. In Chapter 3, the new approach to the analysis of the division with reminder algorithm is presented, based on set theoretic partial orderings (Section 3.3). Thus, the new evidence for (proof of ) Buchbergers result on niteness of the division procedure, regardless of a choice of the leading terms for the next dividing, is obtained. Likewise, in order to investigate Hilbert's original contribution and links with later considerations, we present his proof of the famous Hilbert basis theorem, on the basis of his original papers (section 3.5). In Chapter 4 (page. 41-51), a result from Chapter 3 is used for another new characterization of Gr obner basis, apparently weaker than well-known ones, to be obtained. Namely, it is shown that the bases G = ff1; : : : ; fkg of an ideal I is Gr obner if and only if, for any f from I, the support SuppG(f) is not empty. This condition is (only outwardly) weaker than a typical condition that the leading term of f belongs to SuppG(f). And describe the Gr obner fan of an ideal I and give an algorithm In a special case of two variables. URI: http://hdl.handle.net/123456789/4304 Files in this item: 1
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