Binarne kvadratne forme

master's thesis

• Author: Andreja Rogar
• Mentor: Zrinka Franušić (mentor)
• Committee member: Zrinka Franušić (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Ilja Gogić (član povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-07-17, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Izrazi oblika \[ ax^2+bxy+cy^2, \] gdje su \(a, b, c \in \mathbb{Z}\), nazivaju se binarnim kvadratnim formama i to su zapravo kvadratni homogeni polinomi u dvije varijable. Tijekom povijesti najviše su za proučavane forme oblika \(x^2+y^2\) i \(x^2-dy^2\), dok u današnje vrijeme raste važnost teorije binarnih kvadratnih formi zbog njihove primjene u kriptografiji. U ovom radu su predstavljene osnovne notacije, činjenice i svojstva koja vežemo uz kvadratne forme, poput ekvivalencije formi, njihovog matričnog zapisa, geometrijske interpretacije i diskriminante kvadratnih formi. Nadalje, bavimo se problem reprezentacije cijelog broja nekom kvadratnom formom. Kažemo da forma \(f(x,y)=ax^2+bxy+cy^2\) reprezentira cijeli broj \(n\) ako postoje \(x_{0}, y_{0}\in\mathbb{Z}\) takvi da je \(f(x_{0}, y_{0})=n.\)

Abstract (english)

Expressions such as \[ ax^2+bxy+cy^2, \] where \(a, b, c\in\mathbb{Z}\), are called binary quadratic forms. There are, in fact, quadratic homogeneous polinomials in two variables x and y. Throughout history quadratic forms such as \(x^2+y^2\)and \(x^2-dy^2\) have been studied the most, while today increases the importance of the theory of binary quadratic forms due to their application in cryptography. In this graduate thesis we expose basic notations, facts and properties associated with quadratic forms, such as equivalence of forms, the matrix representation, geometrical interpretations and the discriminant of a form. Furthermore, we deal with representations of integers by binary quadratic forms. We say that the form \(f(x,y)=ax^2+bxy+cy^2\) represents an integer if there exist \(x_{0}, y_{0}\in\mathbb{Z}\) such that \(f(x_{0}, y_{0})=n.\)

• Keywords: binarne kvadratne forme; kriptografija; reprezentacija cijelog broja kvadratnom formom
• Keywords (english): binary quadratic forms; cryptography; representations of integers by binary quadratic forms
• Language: croatian
• URN:NBN: urn:nbn:hr:217:978889
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2018-08-30 12:21:11