Kombinatorični izrek o ničlah

diplomsko delo

Iza Hanžek Šušteršič (Author), Boštjan Kuzman (Mentor)

Abstract

V diplomski nalogi najprej obravnavamo Hilbertov izrek o ničlah (Nullstellensatz) in ga predstavimo z nekaj primeri. Bistvo naloge je Alonov kombinatorični izrek o ničlah, za katerega zahtevamo nekoliko strožje pogoje in dobimo tudi močnejši zaključek. Ogledamo si tudi njegovo posledico, ki se izkaže za močno orodje pri dokazovanju nekaterih znanih izrekov. Tako kombinatorični izrek o ničlah kot njegova posledica sta v nalogi v celoti dokazana. V nadaljevanju si ogledamo nekaj primerov uporabe kombinatoričnega izreka pri dokazovanju že znanih izrekov iz različnih področij matematike, kot sta izrek Chevalleya in Warninga o skupnih ničlah končne družine polinomov, izrek Cauchyja in Davenporta o velikosti vsote podmnožic Zp ter še nekaj drugih zgledov iz geometrije in teorije grafov.

Keywords

kombinatorični izrek o ničlah;polinomi;kombinatorika;

Data

Language:	Slovenian
Year of publishing:	2018
Typology:	2.11 - Undergraduate Thesis
Organization:	UL PEF - Faculty of Education
Publisher:	[I. Hanžek Šušteršič]
UDC:	51(043.2)
COBISS:	12150857
Views:	453
Downloads:	89
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Combinatorial nullstellensatz
Secondary abstract:	In this diploma thesis we first look at Hilbert Nullstellensatz, along with some examples. The main focus of this work, however, is combinatorial nullstellensatz by Alon, which requires stricter conditions and provides a stronger result. We also look at its corollary, which turns out to be a powerful tool when proving some already known theorems. We present detailed proof of both the Combinatorial Nullstellensatz and its corollary in this work. Afterwards, we look at some cases where we can use the combinatorial nullstellensatz to prove already known theorems from different fields of mathematics, such as the Chevalley-Warning theorem of common zeros of a family of polynomials, the Cauchy-Davenport theorem of cardinality of two nonempty subsets of Zp, and some other examples from geometry and graph theory.
Secondary keywords:	mathematics;matematika;
File type:	application/pdf
Type (COBISS):	Bachelor thesis/paper
Thesis comment:	Univ. v Ljubljani, Pedagoška fak., Dvopredmetni učitelj
Pages:	24 str.
ID:	10973411