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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. |
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