Kolobarji polinomov in Gröbnerjeve baze

diplomsko delo

Maja Nemec (Author), Dominik Benkovič (Mentor)

Abstract

V diplomskem delu najprej predstavimo osnovne definicije teorije kolobarjev, ki jih potrebujemo v diplomskem delu. Nato seznanimo s pojmi kolobarjev polinomov ene in več spremenljivk ter noetherskimi kolobarji. Obravnavamo predvsem lastnosti, ki jih ima komutativni kolobar polinomov F[x_1%x_2%%,x_{n}], kjer je F polje. V preostalih poglavjih se posvetimo študiju Gröbnerjevih baz. Gre za končne množice generatorjev posameznih idealov kolobarja F[x_1%x_2%%,x_{n}]. Nato opišemo algoritem splošnega deljenja polinomov, kjer so polinomi iz F[x_1%x_2%%,x_{n}], ki ga rabimo v nadaljevanju. Nadalje povemo nekaj o uporabnosti Gröbnerjevih baz. Med drugim dokažemo Buchbergerjev kriterij, ki igra ključno vlogo pri Buchbergerjevem algoritmu. S pomočjo omenjenega algoritma lahko poiščemo Gröbnerjevo bazo poljubnega danega ideala. Definiramo pojma minimalne in reducirane Gröbnerjeve baze in pogledamo kako iz Gröbnerjeve baze dobimo omenjeni bazi. Teorija Gröbnerjevih baz se izkaže za zelo koristno pri reševanju algebraičnih enačb, saj služi kot osnova pri reševanju sistemov enačb, kjer nastopajo nelinearni polinomi. Zato si na koncu ogledamo eliminacijsko teorijo, ki pove, kako iz nekega sistema enačb dobiti nove enačbe, ki ne vsebujejo vseh prvotnih spremenljivk.

Keywords

matematika;polinomi;kolobarji;Gröbnerjeve baze;noetherski kolobar;algoritmi;diplomska dela;

Data

Language:	Slovenian
Year of publishing:	2010
Source:	Maribor
Typology:	2.11 - Undergraduate Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[M. Nemec]
UDC:	51(043.2)
COBISS:	17885192
Views:	2269
Downloads:	210
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	POLYNOMIAL RINGS AND GRd6BNER BASES
Secondary abstract:	In the beginning of the graduation thesis we present basic definitions of ring theory that are needed through the thesis. Next, we present the concept of a polynomial ring in one or more variables and also noetherian rings. We study properties of a commutative polynomial ring F[x%,x%,%,x_{n}], where F is a field. Next, we focus to the study of Gröbner bases. These are finite sets of generators of ideals of F[x%,x%,%,x_{n}]. Next, we describe an algorithm for a general polynomial division for polynomials in F[x%,x%,%,x_{n}], which is needed in the sequel. We mention the usefulness of Gröbner bases and prove the Buchberger's criterion which plays a crucial role in Buchberger's algorithm. With the help of the algorithm mentioned above we can find a Gröbner basis for any given ideal. We also define the notions of a minimal and reduced Gröbner basis and we show how to get both mentioned bases from a Gröbner basis. The theory of Gröbner bases turns out to be very useful in solving algebraic equations for it is used as the basis in solving the systems of equations where polynomials are nonlinear. Thus at the end we take a look at the elimination theory that tells us how to find new equations from a system of equations that do not involve some of the variables.
Secondary keywords:	polynomial ring;noetherian ring;Gröbner basis;Buchberger's algorithm;
URN:	URN:SI:UM:
Type (COBISS):	Undergraduate thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	54 f.
Keywords (UDC):	mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika;
ID:	18793